A Nonhomogeneous Fractional p-Kirchhoff Type Problem Involving Critical Exponent in ℝN

Mingqi Xiang; Binlin Zhang; Xia Zhang

doi:10.1515/ans-2016-6002

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A Nonhomogeneous Fractional p-Kirchhoff Type Problem Involving Critical Exponent in ℝ^N

Mingqi Xiang , Binlin Zhang and Xia Zhang

Published/Copyright: November 16, 2016

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From the journal Advanced Nonlinear Studies Volume 17 Issue 3

Abstract

This paper concerns itself with the nonexistence and multiplicity of solutions for the following fractional Kirchhoff-type problem involving the critical Sobolev exponent:

$[ a + b ⁢ ( ∬ ℝ 2 ⁢ N | u ⁢ ( x ) - u ⁢ ( y ) | p | x - y | N + p ⁢ s ⁢ 𝑑 x ⁢ 𝑑 y ) θ - 1 ] ⁢ ( - Δ ) p s ⁢ u = | u | p s * - 2 ⁢ u + λ ⁢ f ⁢ ( x ) in ⁢ ℝ N ,$

where $a≥0$ , $b>0,θ>1$ , $(-Δ)ps$ is the fractional p-Laplacian with $0<s<1$ and $1<p<N/s$ , $ps*=N⁢p/(N-p⁢s)$ is the critical Sobolev exponent, $λ≥0$ is a parameter, and $f∈Lps*/(ps*-1)⁢(ℝN)∖{0}$ is a nonnegative function. When $λ=0$ , we show that the multiplicity and nonexistence of solutions for the above problem are related with N, θ, s, p, a, and b. When $λ>0$ , by using Ekeland’s variational principle and the mountain pass theorem, we show that there exists $λ**>0$ such that the above problem admits at least two nonnegative solutions for all $λ∈(0,λ**)$ . In the latter case, in order to overcome the loss of compactness, we derive a fractional version of the principle of concentration compactness in the setting of the fractional p-Laplacian.

Keywords: Critical Sobolev Exponent; Principle of Concentration Compactness

MSC 2010: 35R11; 35A15; 47G20

1 Introduction and Main Results

In this paper, we prove the existence and multiplicity of nonnegative solutions of critical elliptic equations involving the fractional p-Laplace operator $(-Δ)ps$ . More precisely, we consider

(1.1)

${ [ a + b ⁢ ( ∬ ℝ 2 ⁢ N | u ⁢ ( x ) - u ⁢ ( y ) | p | x - y | N + p ⁢ s ⁢ 𝑑 x ⁢ 𝑑 y ) θ - 1 ] ⁢ ( - Δ ) p s ⁢ u = | u | p s * - 2 ⁢ u + λ ⁢ f ⁢ ( x ) in ⁢ ℝ N , u ∈ D s , p ⁢ ( ℝ N ) ,$

where $N>s⁢p$ with $s∈(0,1)$ , $a≥0$ , $b>0$ , $θ>1$ , and $(-Δ)ps$ is the fractional p-Laplacian which (up to normalization factors) may be defined for any $x∈ℝN$ as

$( - Δ ) p s ⁢ φ ⁢ ( x ) = lim ε → 0 ⁡ ∫ ℝ N ∖ B ε ⁢ ( x ) | φ ⁢ ( x ) - φ ⁢ ( y ) | p - 2 ⁢ ( φ ⁢ ( x ) - φ ⁢ ( y ) ) | x - y | N + p ⁢ s ⁢ 𝑑 y$

along any $φ∈C0∞⁢(ℝN)$ , where $Bε⁢(x)$ denotes the ball in $ℝN$ centered at x with radius ε. For more details about the fractional p-Laplacian we refer to [15, 40] and the references therein. Here $Ds,p⁢(ℝN)$ denotes the completion of $C0∞⁢(ℝN)$ with respect to the norm

$∥ u ∥ = ( ∬ ℝ 2 ⁢ N | u ⁢ ( x ) - u ⁢ ( y ) | p | x - y | N + p ⁢ s ⁢ 𝑑 x ⁢ 𝑑 y ) 1 / p$

for all $u∈C0∞⁢(ℝN)$ , and $(Ds,p⁢(ℝN))′$ is the dual space of $Ds,p⁢(ℝN)$ ; see [6]. Without further mentioning, we always assume that $s∈(0,1)$ , $N>s⁢p$ , $θ>1$ , and f satisfies

$f ≥ 0$ , $f≢0$ and $f∈L(ps*)′⁢(ℝN)$ , where $(ps*)′=ps*/(ps*-1)$ is the conjugate exponent of $ps*$ .

The study on semilinear elliptic equations involving critical exponent begins from the seminal paper by Brézis and Nirenberg [11]. After that many authors were dedicated to investigating all kinds of elliptic equations with critical growth in bounded domain or in the whole space. Recently, the solvability or multiplicity of the Kirchhoff-type equation with critical exponent has been paid much attention by many authors. For instance, in bounded domains we refer to [17, 22, 30]; in the whole space, see [21, 26, 24].

In the last years, great attention has been attracted to the study of fractional and nonlocal problems involving critical nonlinearities. For example, some of the recent contributions on the existence of solutions for critical fractional Laplacian equations in bounded domain are given in [8], where the effects of lower order perturbations are considered. A Brézis–Nirenberg-type result for the nonlocal fractional Laplacian in bounded domain with homogeneous Dirichlet boundary datum is given in [37] by variational techniques; see also [36] for further results. Nonexistence results for nonlocal equations involving critical and supercritical nonlinearities can be found in [35]. A multiplicity result for fractional Laplacian problems in $ℝN$ is obtained in [6] by using the mountain pass theorem and the direct method in variational methods, where one of two superlinear nonlinearities could be critical or even supercritical. It is worth mentioning that the interest in nonlocal fractional problems goes beyond the mathematical curiosity. Indeed, this type of operator arises in a quite natural way in many different applications such as continuum mechanics, phase transition phenomena, population dynamics, minimal surfaces and game theory; see for example [4, 12, 15, 23] and the references therein. The literature on nonlocal operators and their applications is quite large, here we just quote a few; see [29, 27, 28, 41] and the references therein. For the basic properties of fractional Sobolev spaces we refer the readers to [15].

Recently in [18], Fiscella and Valdinoci first proposed a stationary Kirchhoff variational equation which models the nonlocal aspect of the tension arising from nonlocal measurements of the fractional length of the string. More precisely, they established a model given by the following formulation:

(1.2)

${ M ⁢ ( ∬ ℝ 2 ⁢ N | u ⁢ ( x ) - u ⁢ ( y ) | 2 | x - y | N + 2 ⁢ s ⁢ 𝑑 x ⁢ 𝑑 y ) ⁢ ( - Δ ) s ⁢ u = λ ⁢ f ⁢ ( x , u ) + | u | 2 s * - 2 ⁢ u in ⁢ Ω , u = 0 in ⁢ ℝ N ∖ Ω ,$

where $M⁢(t)=a+b⁢t$ for all $t≥0$ , with $a>0$ , $b≥0$ . Note that if M is of this type, problem (1.2) is called non-degenerate if $a>0$ and $b≥0$ , while it is named degenerate if $a=0$ and $b>0$ ; see [34] for some physical explanations about non-degenerate Kirchhoff problems. Hence problem (1.2) is non-degenerate. For some motivation for the physical background of the fractional Kirchhoff model we refer to [18, Appendix A]. In [38], Xiang, Zhang and Ferrara investigated the existence of solutions for Kirchhoff-type problems involving the fractional p-Laplacian via variational methods, where the nonlinearity is subcritical and the Kirchhoff function is non-degenerate. By the use of the mountain pass theorem and Ekeland’s variational principle, the authors in [39] studied the multiplicity of solutions to a nonhomogeneous Kirchhoff-type problem driven by the fractional p-Laplacian, where the nonlinearity is convex-concave and the Kirchhoff function is degenerate. By the use of the same methods as in [39], Pucci, Xiang and Zhang obtained in [33] the existence of multiple solutions for the nonhomogeneous fractional p-Laplacian equations of Schrödinger–Kirchhoff type in the whole space. Indeed, the fractional Kirchhoff problems have been extensively studied in recent years; for instance, see [32] about non-degenerate Kirchhoff-type problems and [5, 34] about degenerate Kirchhoff-type problems for the recent advances in this direction. However, to our best knowledge, there is no result in the literature on problem (1.1). Therefore, in the present paper we are interested in the nonexistence and multiplicity of solutions for problem (1.1) involving the fractional p-Laplacian in $ℝN$ . There is no doubt that we encounter serious difficulties because of the lack of compactness and of the nonlocal nature of the p-fractional Laplacian. For this purpose, we shall extend the principle of concentrate compactness of Lions in [25] to the context of fractional p-Laplacian in the whole space. We would like to stress that the extension from the case $p=2$ to the case $1<p<∞$ is not trivial.

Note that if $a=1$ , $b=0$ and $λ=0$ , problem (1.1) reduces to the following fractional p-Laplacian problem:

(1.3)

${ ( - Δ ) p s ⁢ u = | u | p s * - 2 ⁢ u , x ∈ ℝ N , u ∈ D s , p ⁢ ( ℝ N ) .$

Let

$S = inf u ∈ D s , p ⁢ ( ℝ N ) ∖ { 0 } ⁡ ∥ u ∥ p | u | p s * p ,$

which is positive by the fractional Sobolev inequality. Here and throughout this paper, we shortly denote by $|⋅|q$ the norm of $Lq⁢(ℝN)$ for any $q∈(1,∞)$ . Very recently, Brasco, Mosconi and Squassina obtained in [10] that there exists a radially symmetric nonnegative decreasing minimizer $U=U⁢(r)$ for S. The authors also showed that U weakly solves (1.3) and satisfies

(1.4)

$∥ U ∥ p = | U | p s * p s * = S N / s ⁢ p .$

Moreover, for any $ε>0$ the function

(1.5)

$U ε ⁢ ( x ) = 1 ε ( N - s ⁢ p ) / p ⁢ U ⁢ ( | x | / ε )$

is also a minimizer for S satisfying (1.3) and (1.4). We first give the definition of solutions for problem (1.1).

Definition 1.1.

We say that $u∈Ds,p⁢(ℝN)$ is a (weak) solution of equation (1.1) if

$( a + b ⁢ ∥ u ∥ ( θ - 1 ) ⁢ p ) ⁢ ∬ ℝ 2 ⁢ N | u ⁢ ( x ) - u ⁢ ( y ) | p - 2 ⁢ ( u ⁢ ( x ) - u ⁢ ( y ) ) | x - y | N + p ⁢ s ⁢ ( φ ⁢ ( x ) - φ ⁢ ( y ) ) ⁢ 𝑑 x ⁢ 𝑑 y$

$= ∫ ℝ N | u ⁢ ( x ) | p s * - 2 ⁢ u ⁢ ( x ) ⁢ φ ⁢ ( x ) ⁢ 𝑑 x + λ ⁢ ∫ ℝ N f ⁢ ( x ) ⁢ φ ⁢ ( x ) ⁢ 𝑑 x ,$

for any $φ∈Ds,p⁢(ℝN)$

Theorem 1.2.

Assume $λ=0$ and $θ>1$ . For $Uε$ given by (1.5) the following conclusions hold:

If $θ = N / ( N - s ⁢ p )$ , $a>0$ and $b<S-θ$ , then problem (1.1) has infinitely many nonnegative solutions and these solutions are

$( a 1 - b ⁢ S θ ) 1 / ( θ - 1 ) ⁢ p ⁢ U ε$

for all $ε > 0$ .

If $θ = ( N + s ⁢ p ) / ( N - s ⁢ p )$ and $a , b > 0$ satisfying $4 ⁢ a ⁢ b ⁢ S θ + 1 < 1$ , then problem ( 1.1 ) has infinitely many solutions and these solutions are

$( 1 - 1 - 4 ⁢ a ⁢ b ⁢ S θ + 1 2 ⁢ b ⁢ S θ + 1 ) 1 / ( p s * - p ) ⁢ U ε$

and

$( 1 + 1 - 4 ⁢ a ⁢ b ⁢ S θ + 1 2 ⁢ b ⁢ S θ + 1 ) 1 / ( p s * - p ) ⁢ U ε$

for all $ε > 0$ .
If $θ ≠ N / ( N - s ⁢ p )$ , $a=0$ and $b>0$ , then problem (1.1) has infinitely many nonnegative solutions and these solutions are

$( b ⁢ S N ⁢ ( θ - 1 ) / p ⁢ s ) - 1 / ( θ ⁢ p - p s * ) ⁢ U ε for all ⁢ ε > 0 .$

If $θ > N / ( N - s ⁢ p )$ and $a , b > 0$ satisfying

(1.6) $a θ ⁢ p - p s * ⁢ ( b ⁢ S N ⁢ ( θ - 1 ) p ⁢ s ) p s * - p = ( θ ⁢ p - p s * ) θ ⁢ p - p s * ⁢ ( p s * - p ) p s * - p ( ( θ - 1 ) ⁢ p ) p ⁢ ( θ - 1 ) ,$

then problem ( 1.1 ) has infinitely many nonnegative solutions and these solutions are $μ 0 1 / ( p s * - p ) ⁢ U ε$ for all $ε > 0$ , where $μ 0 = p s * - p b ⁢ p ⁢ ( θ - 1 ) ⁢ S N ⁢ ( θ - 1 ) / ( p ⁢ s )$ .

If $θ > N / ( N - s ⁢ p )$ and $a , b > 0$ satisfying

(1.7) $a θ ⁢ p - p s * ⁢ ( b ⁢ S N ⁢ ( θ - 1 ) p ⁢ s ) p s * - p < ( θ ⁢ p - p s * ) θ ⁢ p - p s * ⁢ ( p s * - p ) p s * - p ( ( θ - 1 ) ⁢ p ) p ⁢ ( θ - 1 ) ,$

then there exist $μ 1 ∈ ( 0 , μ 0 )$ and $μ 2 ∈ ( μ 0 , ∞ )$ such that $μ 1 1 / ( p s * - p ) ⁢ U ε$ and $μ 2 1 / ( p s * - p ) ⁢ U ε$ are solutions of problem ( 1.1 ) for all $ε > 0$ .
If $θ < N / ( N - s ⁢ p )$ , $a>0$ and $b>0$ , then there exists $μ0∈(0,∞)$ such that $μ01/(ps*-p)⁢Uε$ are solutions of problem (1.1), where $ε>0$ .

Theorem 1.3.

Suppose that $λ=0$ . Then problem (1.1) has no nontrivial solution, under one of the following hypotheses:

$θ = N / ( N - s ⁢ p )$ , $a=0$ and $b>S-θ$ .
$θ = N / ( N - s ⁢ p )$ , $a>0$ and $b≥S-θ$ .
$θ = 2$ and $a , b > 0$ satisfy

(1.8) $( p s * p - 1 ) ⁢ ( p ⁢ a 2 ⁢ p - p s * ) ( p s * - 2 ⁢ p ) / ( p s * - p ) ⁢ S - p s * / ( p s * - p ) < b .$

Remark 1.4.

(i) The proof of Theorem 1.2 relies on the solutions of problem (1.3). As stated in [10], problem (1.3) has infinitely many nonnegative solutions $Uε$ . Thus we obtain also that problem (1.1) with $λ=0$ has infinitely many nonnegative solutions, under some suitable assumptions on a and b.

(ii) By Theorem 1.2 and Theorem 1.3, it is easy to see that the existence of solutions for problem (1.1) is related with λ and a, b. When $θ=2$ and $N/(N-s⁢p)<2$ , we obtain by Theorem 1.2 (ii) and Theorem 1.3 (iii) that problem (1.1) with $λ=0$ has infinitely many nonnegative nontrivial solutions if conditions (1.6) or (1.7) hold, and has no nontrivial solution if assumption (1.8) holds.

Next we define

$I λ ⁢ ( u ) = a p ⁢ ∥ u ∥ p + b θ ⁢ p ⁢ ∥ u ∥ θ ⁢ p - 1 p s * ⁢ ∫ ℝ N ( u + ) p s * ⁢ 𝑑 x - λ ⁢ ∫ ℝ N f ⁢ ( x ) ⁢ u ⁢ 𝑑 x ,$

associated to problem (1.1), for all $u∈Ds,p⁢(ℝN)$ , where $u+=max⁡{u,0}$ . Note that the condition $f∈(Ds,p⁢(ℝN))′$ implies that $Iλ∈C1⁢(Ds,p⁢(ℝN))$ and

$〈 I λ ′ ⁢ ( u ) , φ 〉 = ( a + b ⁢ ∥ u ∥ ( θ - 1 ) ⁢ p ) ⁢ ∬ ℝ 2 ⁢ N | u ⁢ ( x ) - u ⁢ ( y ) | p - 2 ⁢ ( u ⁢ ( x ) - u ⁢ ( y ) ) ⁢ ( φ ⁢ ( x ) - φ ⁢ ( y ) ) | x - y | N + p ⁢ s ⁢ 𝑑 x ⁢ 𝑑 y$

$- ∫ ℝ N ( u + ) p s * - 1 ⁢ φ ⁢ 𝑑 x - λ ⁢ ∫ ℝ N f ⁢ φ ⁢ 𝑑 x$

for all $φ∈Ds,p⁢(ℝN)$ . Hence a critical point of functional $Iλ$ is a weak solution of problem (1.1).

Now we are in a position to state the third result of our paper as follows.

Theorem 1.5.

Assume that $θ<N/(N-s⁢p)$ , $a≥0$ and $b>0$ , or $θ=N/(N-s⁢p)$ , $a>0$ and $0<b<S-θ$ , or $θ=(N-s⁢p/2)/(N-s⁢p)$ , $a>0$ and $b>0$ . Suppose that (f) holds. Then there exists a constants $λ*>0$ such that for any $λ∈(0,λ*)$ problem (1.1) has a nonnegative solution $u1$ with $Iλ⁢(u1)<0$ . Moreover, there exists $λ**∈(0,λ*]$ such that for any $λ∈(0,λ**)$ problem (1.1) has another nonnegative solution $u2$ with $Iλ⁢(u2)>0$ .

Remark 1.6.

If $θ=2$ and $p=2$ , then by Theorem 1.5 we obtain the existence of two nonnegative solutions of the following problem:

${ ( a + b ⁢ ∬ ℝ 2 ⁢ N | u ⁢ ( x ) - u ⁢ ( y ) | 2 | x - y | N + 2 ⁢ s ⁢ 𝑑 x ⁢ 𝑑 y ) ⁢ ( - Δ ) s ⁢ u = | u | 2 s * - 2 ⁢ u + λ ⁢ f ⁢ ( x ) in ⁢ ℝ N , u ∈ D s ⁢ ( ℝ N ) ,$

where $N<4⁢s$ . In this case, the result is obtained under the condition $2⁢s<N<4⁢s$ . Thus, our result does not contain the interesting case $s=1/2$ . However, for the case $s=1/2$ , $p=2$ and $θ=1+1/2$ , we obtain the existence of two nonnegative solutions for the problem

${ ( a + b ⁢ ∫ ℝ 2 ∫ ℝ 2 | u ⁢ ( x ) - u ⁢ ( y ) | 2 | x - y | 3 ⁢ 𝑑 x ⁢ 𝑑 y ) ⁢ ( - Δ ) 1 2 ⁢ u = u 3 + λ ⁢ f ⁢ ( x ) in ⁢ ℝ 2 , u ∈ D s ⁢ ( ℝ 2 ) ,$

where $a≥0$ and $b>0$ , or the problem

${ ( a + b ⁢ ∫ ℝ 3 ∫ ℝ 3 | u ⁢ ( x ) - u ⁢ ( y ) | 2 | x - y | 4 ⁢ 𝑑 x ⁢ 𝑑 y ) ⁢ ( - Δ ) 1 2 ⁢ u = u 2 + λ ⁢ f ⁢ ( x ) in ⁢ ℝ 3 , u ∈ D s ⁢ ( ℝ 3 ) ,$

where $a>0$ , $0<b<S-3/2$ . Now we consider the case $s=1/2$ , $p=2$ and $θ=1+ε$ with $ε>0$ . Then Theorem 1.5 shows that the following problem has at least two nonnegative solutions:

${ ( a + b ⁢ ( ∬ ℝ 2 ⁢ N | u ⁢ ( x ) - u ⁢ ( y ) | 2 | x - y | N + 1 ⁢ 𝑑 x ⁢ 𝑑 y ) ε ) ⁢ ( - Δ ) 1 2 ⁢ u = | u | 2 N - 1 ⁢ u + λ ⁢ f ⁢ ( x ) in ⁢ ℝ N , u ∈ D s ⁢ ( ℝ N ) ,$

where $1<N<1+1/ε$ , $a>0$ and $b>0$ . This means that we obtain a large range of N if $ε>0$ is small enough.

Finally, let us simply describe the main approach to obtain Theorem 1.5. To show the existence of at least two critical points of the energy functional, we shall use Ekeland’s variational principle [16], and the mountain pass theorem of Ambrosetti and Rabinowitz [3] without the Palais–Smale condition. Using Ekeland’s variational principle, we obtain a solution $u1$ with $Iλ⁢(u1)<0$ , and by the mountain pass theorem we prove the existence of a second solution $u2$ with $Iλ⁢(u2)>0$ . Techniques for finding the solutions $u1$ and $u2$ are borrowed from Cao, Li and Zhou [13]. Then we combine these techniques with arguments developed by Chabrowski [14], Alves, Gongcalves and Miyagaki [2], and Alves [1] to prove Theorem 1.5.

This paper is organized as follows: Section 2 is devoted to proving the fractional version of the principle of concentration compactness of Lions [25] in fractional Sobolev spaces $Ds,p⁢(ℝN)$ . In Section 3, we give the proofs of Theorems 1.2 and 1.3 in the case $λ=0$ . In Section 4, using Ekeland’s variational principle and the mountain pass theorem, we establish the existence of two nonnegative solutions for problem (1.1) with a suitable range of the positive parameter λ.

2 Fractional Version of the Principle of Concentration Compactness

In this section, in order to overcome the lack of some compactness, we turn to investigate the fractional version of the principle of concentration compactness of Lions [25] in fractional Sobolev spaces $Ds,p⁢(ℝN)$ , which will be used in Section 4.

In [25], Lions established the principle of concentration compactness in classical Sobolev spaces, and then the principle of concentration compactness was well used to solve some elliptic problems involving critical exponent; see also [19] for the principle of concentration compactness in variable exponent Sobolev spaces. In [31], the authors established the principle of concentration compactness in fractional Sobolev spaces $Ds⁢(ℝN)$ by using profile decomposition; see also [9] for its applications. However, the principle of concentration compactness in [31] seems not to be convenient to solve our problem because of the different way of defining the fractional Laplacian and the different working spaces. To this end, we will establish the principle of concentration compactness in $Ds,p⁢(ℝN)$ , which can be regarded as a fractional counterpart of the principle of concentration compactness in the classical Sobolev space $W1,p⁢(ℝN)$ . Let

$C c ⁢ ( ℝ N ) = { u ∈ C ⁢ ( ℝ N ) : supp ⁢ ( u ) ⁢ is a compact subset of ⁢ ℝ N }$

and denote by $C0⁢(ℝN)$ the closure of $Cc⁢(ℝN)$ with respect to the norm $|η|∞=supx∈ℝN⁡|η⁢(x)|$ . As is well known, a finite measure on $ℝN$ is a continuous linear functional on $C0⁢(ℝN)$ . For a measure μ we give the norm

$∥ μ ∥ = sup C 0 ⁢ ( ℝ N ) , | η | ∞ = 1 ⁡ | ( μ , η ) | ,$

where $(μ,η)=∫ℝNη⁢𝑑μ$ .

Definition 2.1.

Let $ℳ⁢(ℝN)$ denote the finite nonnegative Borel measure space on $ℝN$ . For any $μ∈ℳ⁢(ℝN)$ the equation $μ⁢(ℝN)=∥μ∥$ holds. We say that $μn⇀μ$ weakly * in $ℳ⁢(ℝN)$ if $(μn,η)→(μ,η)$ holds for all $η∈C0⁢(ℝN)$ as $n→∞$ .

Theorem 2.2.

Let ${un}n⊂Ds,p⁢(RN)$ with upper bound $C>0$ for all $n≥1$ and

${ u n ⇀ u ⁢ weakly in ⁢ D s , p ⁢ ( ℝ N ) , ∫ ℝ N | u n ⁢ ( x ) - u n ⁢ ( y ) | p | x - y | N + s ⁢ p ⁢ 𝑑 y ⇀ μ ⁢ weakly * in ⁢ ℳ ⁢ ( ℝ N ) , | u n ⁢ ( x ) | p s * ⇀ ν ⁢ weakly * in ⁢ ℳ ⁢ ( ℝ N ) .$

Then

$μ = ∫ ℝ N | u ⁢ ( x ) - u ⁢ ( y ) | p | x - y | N + s ⁢ p ⁢ 𝑑 y + ∑ j ∈ J μ j ⁢ δ x j + μ ~ ,$

$⁢ μ ⁢ ( ℝ N ) ≤ C p ,$

$ν = | u | p s * + ∑ j ∈ J v j ⁢ δ x j ,$

$⁢ ν ⁢ ( ℝ N ) ≤ S p s * ⁢ C p ,$

where J is at most countable, sequences ${μj}j,{νj}j⊂R0+$ , ${xj}j⊂RN$ , $δxj$ is the Dirac mass centered at $xj$ , $μ~$ is a non-atomic measure,

$ν ⁢ ( ℝ N ) ≤ S - p s * / p ⁢ μ ⁢ ( ℝ N ) p s * / p , ν j ≤ S - p s * / p ⁢ μ j p s * / p for all ⁢ j ∈ J ,$

and $S>0$ is the best constant of $Ds,p⁢(RN)↪Lps*⁢(RN)$ .

In what follows, we write $K⁢(x-y)=|x-y|-N-s⁢p$ for a notational convenience.

Lemma 2.3.

Assume that ${un}n⊂Ds,p⁢(RN)$ is the sequence given by Theorem 2.2, let $x0∈RN$ fixed and let $φ∈C0∞⁢(RN)$ such that $0≤φ≤1$ , $φ≡1$ in $B⁢(0,1)$ , $φ≡0$ in $RN∖B⁢(0,2)$ , and $|∇⁡φ|≤2$ . Set $φε,0⁢(x)=φ⁢((x-x0)/ε)$ for all $x∈RN$ . Then

$lim ε → 0 ⁡ lim sup n → ∞ ⁡ ∬ ℝ 2 ⁢ N | u n ⁢ ( x ) | p ⁢ | φ ε , 0 ⁢ ( x ) - φ ε , 0 ⁢ ( y ) | p ⁢ K ⁢ ( x - y ) ⁢ 𝑑 x ⁢ 𝑑 y = 0 .$

Proof.

We first observe that

$ℝ N × ℝ N = ( ( ℝ N ∖ B ⁢ ( x 0 , 2 ⁢ ε ) ) ∪ B ⁢ ( x 0 , 2 ⁢ ε ) ) × ( ( ℝ N ∖ B ⁢ ( x 0 , 2 ⁢ ε ) ) ∪ B ⁢ ( x 0 , 2 ⁢ ε ) )$

$= ( ( ℝ N ∖ B ⁢ ( x 0 , 2 ⁢ ε ) ) × ( ℝ N ∖ B ⁢ ( x 0 , 2 ⁢ ε ) ) ) ∪ ( B ⁢ ( x 0 , 2 ⁢ ε ) × ℝ N ) ∪ ( ( ℝ N ∖ B ⁢ ( x 0 , 2 ⁢ ε ) ) × B ⁢ ( x 0 , 2 ⁢ ε ) ) .$

Thus we divide the whole proof into the following three cases.

Case 1. If $(x,y)∈(ℝN∖B⁢(x0,2⁢ε))×(ℝN∖B⁢(x0,2⁢ε))$ , then $φε,0⁢(x)=φε,0⁢(y)=0$ .

Case 2. If $(x,y)∈B⁢(x0,2⁢ε)×ℝN$ and $|x-y|≤ε$ , then $|y-x0|≤|x-y|+|x-x0|≤3⁢ε$ , which implies that

$∫ B ⁢ ( x 0 , 2 ⁢ ε ) 𝑑 x ⁢ ∫ { y ∈ ℝ N : | x - y | ≤ ε } | u n ⁢ ( x ) | p ⁢ | φ ε , 0 ⁢ ( x ) - φ ε , 0 ⁢ ( y ) | p ⁢ K ⁢ ( x - y ) ⁢ 𝑑 y$

$= ∫ B ⁢ ( x 0 , 2 ⁢ ε ) 𝑑 x ⁢ ∫ { y ∈ ℝ N : | x - y | ≤ ε } | u n ⁢ ( x ) | p ⁢ | ∇ ⁡ φ ε , 0 ⁢ ( ξ ) | p ⁢ | ( x - y ) / ε | p ⁢ K ⁢ ( x - y ) ⁢ 𝑑 y$

$≤ C ⁢ ε - p ⁢ ∫ B ⁢ ( x 0 , 2 ⁢ ε ) 𝑑 x ⁢ ∫ { y ∈ ℝ N : | x - y | ≤ ε } | u n ⁢ ( x ) | p | x - y | N + p ⁢ s - p ⁢ 𝑑 y$

$≤ C ⁢ ε - p ⁢ ∫ B ⁢ ( x 0 , 2 ⁢ ε ) 𝑑 x ⁢ ∫ 0 ε | u n ⁢ ( x ) | p ⁢ r p - 1 - p ⁢ s ⁢ 𝑑 r ≤ C ⁢ ε - p ⁢ s ⁢ ∫ B ⁢ ( x 0 , 2 ⁢ ε ) | u n ⁢ ( x ) | p ⁢ 𝑑 x ,$

where $ξ=(y-x0)/ε+τ⁢(x-x0)/ε$ and $τ∈(0,1)$ . If $(x,y)∈B⁢(x0,2⁢ε)×ℝN$ and $|x-y|>ε$ , we have

$∫ B ⁢ ( x 0 , 2 ⁢ ε ) 𝑑 x ⁢ ∫ { y ∈ ℝ N : | x - y | > ε } | u n ⁢ ( x ) | p ⁢ | φ ε , 0 ⁢ ( x ) - φ ε , 0 ⁢ ( y ) | p ⁢ K ⁢ ( x - y ) ⁢ 𝑑 y$

$≤ C ⁢ ∫ B ⁢ ( x 0 , 2 ⁢ ε ) 𝑑 x ⁢ ∫ { y ∈ ℝ N : | x - y | > ε } | u n ⁢ ( x ) | p | x - y | N + p ⁢ s ⁢ 𝑑 y$

$≤ C ⁢ ε - p ⁢ s ⁢ ∫ B ⁢ ( x 0 , 2 ⁢ ε ) | u n ⁢ ( x ) | p ⁢ 𝑑 x .$

Case 3. If $(x,y)∈(ℝN∖B⁢(x0,2⁢ε))×B⁢(x0,2⁢ε)$ and $|x-y|≤ε$ , then $|x-x0|≤3⁢ε$ . Thus,

$∫ ℝ N ∖ B ⁢ ( x 0 , 2 ⁢ ε ) 𝑑 x ⁢ ∫ { y ∈ B ⁢ ( x 0 , 2 ⁢ ε ) : | x - y | ≤ ε } | u n ⁢ ( x ) | p ⁢ | φ ε , 0 ⁢ ( x ) - φ ε , 0 ⁢ ( y ) | p ⁢ K ⁢ ( x - y ) ⁢ 𝑑 y$

$≤ C ⁢ ε - p ⁢ ∫ B ⁢ ( x 0 , 3 ⁢ ε ) 𝑑 x ⁢ ∫ { y ∈ B ⁢ ( x 0 , 2 ⁢ ε ) : | x - y | ≤ ε } | u n ⁢ ( x ) | p | x - y | N + p ⁢ s - p ⁢ 𝑑 y$

$≤ C ⁢ ε - p ⁢ ∫ B ⁢ ( x 0 , 3 ⁢ ε ) 𝑑 x ⁢ ∫ { z ∈ ℝ N : | z | ≤ ε } | u n ⁢ ( x ) | p | z | N + p ⁢ s - p ⁢ 𝑑 z$

$≤ C ⁢ ε - p ⁢ s ⁢ ∫ B ⁢ ( x 0 , 3 ⁢ ε ) | u n ⁢ ( x ) | p ⁢ 𝑑 x .$

If $(x,y)∈(ℝN∖B⁢(x0,2⁢ε))×B⁢(x0,2⁢ε)$ and $|x-y|>ε$ , we observe that

$( ℝ N ∖ B ⁢ ( x 0 , 2 ⁢ ε ) ) × B ⁢ ( x 0 , 2 ⁢ ε ) ⊂ ( B ⁢ ( x 0 , k ⁢ ε ) × B ⁢ ( x 0 , 2 ⁢ ε ) ) ∪ ( ( ℝ N ∖ B ⁢ ( x 0 , k ⁢ ε ) ) × B ⁢ ( x 0 , 2 ⁢ ε ) )$

for all $k>4$ . Thus we get

$∫ B ⁢ ( x 0 , k ⁢ ε ) 𝑑 x ⁢ ∫ { y ∈ B ⁢ ( x 0 , 2 ⁢ ε ) : | x - y | > ε } | u n ⁢ ( x ) | p ⁢ | φ ε , 0 ⁢ ( x ) - φ ε , 0 ⁢ ( y ) | p ⁢ K ⁢ ( x - y ) ⁢ 𝑑 y$

$≤ C ⁢ ∫ B ⁢ ( x 0 , k ⁢ ε ) 𝑑 x ⁢ ∫ { y ∈ B ⁢ ( x 0 , 2 ⁢ ε ) : | x - y | > ε } | u n ⁢ ( x ) | p | x - y | N + p ⁢ s ⁢ 𝑑 y$

$≤ C ⁢ ∫ B ⁢ ( x 0 , k ⁢ ε ) 𝑑 x ⁢ ∫ { z ∈ ℝ N : | z | > ε } | u n ⁢ ( x ) | p | z | N + p ⁢ s ⁢ 𝑑 z$

$≤ C ⁢ ε - p ⁢ s ⁢ ∫ B ⁢ ( x 0 , k ⁢ ε ) | u n ⁢ ( x ) | p ⁢ 𝑑 x .$

Moreover, if $(x,y)∈(ℝN∖B⁢(x0,k⁢ε))×B⁢(x0,2⁢ε)$ , then

$| x - y | ≥ | x - x 0 | - | y - y 0 | ≥ | x - x 0 | 2 + k 2 ⁢ ε - 2 ⁢ ε > | x - x 0 | 2 ,$

which implies that

$∫ ℝ N ∖ B ⁢ ( x 0 , k ⁢ ε ) 𝑑 x ⁢ ∫ { y ∈ B ⁢ ( x 0 , 2 ⁢ ε ) : | x - y | > ε } | u n ⁢ ( x ) | p ⁢ | φ ε , 0 ⁢ ( x ) - φ ε , 0 ⁢ ( y ) | p ⁢ K ⁢ ( x - y ) ⁢ 𝑑 y$

$≤ C ⁢ ∫ ℝ N ∖ B ⁢ ( x 0 , k ⁢ ε ) 𝑑 x ⁢ ∫ { y ∈ B ⁢ ( x 0 , 2 ⁢ ε ) : | x - y | > ε } | u n ⁢ ( x ) | p | x - y | N + p ⁢ s ⁢ 𝑑 y$

$≤ C ⁢ ε N ⁢ ∫ ℝ N ∖ B ⁢ ( x 0 , k ⁢ ε ) | u n ⁢ ( x ) | p | x - x 0 | N + p ⁢ s ⁢ 𝑑 x ≤ C ⁢ k - N ⁢ ( ∫ ℝ N ∖ B ⁢ ( x 0 , k ⁢ ε ) | u n ⁢ ( x ) | p s * ⁢ 𝑑 x ) p / p s * .$

Therefore, from Cases 1–3 it follows that

$∬ ℝ 2 ⁢ N | u n ⁢ ( x ) | p ⁢ | φ ε , 0 ⁢ ( x ) - φ ε , 0 ⁢ ( y ) | p ⁢ K ⁢ ( x - y ) ⁢ 𝑑 x ⁢ 𝑑 y$

$= ∬ B ⁢ ( x 0 , 2 ⁢ ε ) × ℝ N | u n ⁢ ( x ) | p ⁢ | φ ε , 0 ⁢ ( x ) - φ ε , 0 ⁢ ( y ) | p ⁢ K ⁢ ( x - y ) ⁢ 𝑑 x ⁢ 𝑑 y$

$+ ∬ ( ℝ N ∖ B ⁢ ( x 0 , 2 ⁢ ε ) ) × B ⁢ ( x 0 , 2 ⁢ ε ) | u n ⁢ ( x ) | p ⁢ | φ ε , 0 ⁢ ( x ) - φ ε , 0 ⁢ ( y ) | p ⁢ K ⁢ ( x - y ) ⁢ 𝑑 x ⁢ 𝑑 y$

$≤ C ⁢ ε - p ⁢ s ⁢ ∫ B ⁢ ( x 0 , 2 ⁢ ε ) | u n ⁢ ( x ) | p ⁢ 𝑑 x + C ⁢ ε - p ⁢ s ⁢ ∫ B ⁢ ( x 0 , 3 ⁢ ε ) | u n ⁢ ( x ) | p ⁢ 𝑑 x + C ⁢ ε - p ⁢ s ⁢ ∫ B ⁢ ( x 0 , k ⁢ ε ) | u n ⁢ ( x ) | p ⁢ 𝑑 x$

$+ C ⁢ k - N ⁢ ( ∫ ℝ N ∖ B ⁢ ( x 0 , k ⁢ ε ) | u n ⁢ ( x ) | p s * ⁢ 𝑑 x ) p / p s *$

$≤ C ⁢ ε - p ⁢ s ⁢ ∫ B ⁢ ( x 0 , k ⁢ ε ) | u n ⁢ ( x ) | p ⁢ 𝑑 x + C ⁢ k - N ⁢ ( ∫ ℝ N ∖ B ⁢ ( x 0 , k ⁢ ε ) | u n ⁢ ( x ) | p s * ⁢ 𝑑 x ) p / p s *$

$≤ C ⁢ ε - p ⁢ s ⁢ ∫ B ⁢ ( x 0 , k ⁢ ε ) | u n ⁢ ( x ) | p ⁢ 𝑑 x + C ⁢ k - N .$

Note that if $un⇀u$ weakly in $Ds,p⁢(ℝN)$ , then $un→u$ in $Llocp⁢(ℝN)$ , which implies that

$C ⁢ ε - p ⁢ s ⁢ ∫ B ⁢ ( x 0 , k ⁢ ε ) | u n ⁢ ( x ) | p ⁢ 𝑑 x + C ⁢ k - N → C ⁢ ε - p ⁢ s ⁢ ∫ B ⁢ ( x 0 , k ⁢ ε ) | u ⁢ ( x ) | p ⁢ 𝑑 x + C ⁢ k - N$

as $n→∞$ . By the Hölder inequality, we obtain

$C ⁢ ε - p ⁢ s ⁢ ∫ B ⁢ ( x 0 , k ⁢ ε ) | u ⁢ ( x ) | p ⁢ 𝑑 x + C ⁢ k - N ≤ C ⁢ ε - p ⁢ s ⁢ ( ∫ B ⁢ ( x 0 , k ⁢ ε ) | u ⁢ ( x ) | p s * ⁢ 𝑑 x ) p / p s * ⁢ ( ∫ B ⁢ ( x 0 , k ⁢ ε ) 𝑑 x ) 1 - p / p s * + C ⁢ k - N$

$≤ C ⁢ k p ⁢ s ⁢ ( ∫ B ⁢ ( x 0 , k ⁢ ε ) | u ⁢ ( x ) | p s * ⁢ 𝑑 x ) p / p s * + C ⁢ k - N → C ⁢ k - N$

as $ε→0$ . Therefore,

$lim ε → 0 ⁡ lim sup n → ∞ ⁡ ∬ ℝ 2 ⁢ N | u n ⁢ ( x ) | p ⁢ | φ ε , 0 ⁢ ( x ) - φ ε , 0 ⁢ ( y ) | p ⁢ K ⁢ ( x - y ) ⁢ 𝑑 x ⁢ 𝑑 y$

$= lim k → ∞ ⁡ lim ε → 0 ⁡ lim sup n → ∞ ⁡ ∬ ℝ 2 ⁢ N | u n ⁢ ( x ) | p ⁢ | φ ε , 0 ⁢ ( x ) - φ ε , 0 ⁢ ( y ) | p ⁢ K ⁢ ( x - y ) ⁢ 𝑑 x ⁢ 𝑑 y = 0 .$

Thus, the proof is finished.∎

Proof of Theorem 2.2.

We divide the proof into four parts. Part 1. We first prove that $μ⁢(ℝN)≤Cp$ and $ν⁢(ℝN)≤(C*)ps*⁢Cp$ . For $R>0$ , take $η∈C0∞⁢(B2⁢R⁢(0))$ satisfying $0≤η≤1$ and $η≡1$ on $BR⁢(0)$ . Then

$∫ ℝ N ∫ ℝ N | u n ⁢ ( x ) - u n ⁢ ( y ) | p ⁢ K ⁢ ( x - y ) ⁢ 𝑑 y ⁢ η ⁢ ( x ) ⁢ 𝑑 x → ∫ ℝ N η ⁢ ( x ) ⁢ 𝑑 μ$

as $n→∞$ . Since $∥un∥≤C$ , we obtain

$∫ ℝ N ∫ ℝ N | u n ⁢ ( x ) - u n ⁢ ( y ) | p ⁢ K ⁢ ( x - y ) ⁢ 𝑑 y ⁢ η ⁢ ( x ) ⁢ 𝑑 x ≤ ∫ ℝ N ∫ ℝ N | u n ⁢ ( x ) - u n ⁢ ( y ) | p ⁢ K ⁢ ( x - y ) ⁢ 𝑑 y ⁢ 𝑑 x ≤ C p .$

Hence, $μ⁢(BR⁢(0))≤∫ℝNη⁢𝑑μ≤Cp$ . Let $R→∞$ . We get that $μ⁢(ℝN)≤Cp$ . Similarly, we have $ν⁢(ℝN)≤S-ps*/p⁢Cps*$ , since $∫ℝN|un|ps*⁢𝑑x≤S-ps*/p⁢Cps*$ by the definition of S and $∥un∥≤C$ .

Part 2. We claim that

$μ = ∫ ℝ N | u ⁢ ( x ) - u ⁢ ( y ) | p | x - y | N + p ⁢ s ⁢ 𝑑 y + ∑ j ∈ J μ j ⁢ δ x j + μ ~ ,$

where ${xj}j⊂ℝN$ , ${μj}j⊂[0,∞)$ , J is an at most countable set, $μ~∈ℳ⁢(ℝN)$ is a nonnegative non-atomic measure and $δxj$ is the Dirac mass at $xj$ . Take $0≤η∈C0⁢(ℝN)$ and set

$ℱ ⁢ ( v ) = ∫ ℝ N ∫ ℝ N | v ⁢ ( x ) - v ⁢ ( y ) | p | x - y | N + p ⁢ s ⁢ 𝑑 y ⁢ η ⁢ ( x ) ⁢ 𝑑 x .$

It is easy to verify that $ℱ$ is a continuously differentiable convex functional on $Ds,p⁢(ℝN)$ . Hence $ℱ$ is weakly lower semicontinuous on $Ds,p⁢(ℝN)$ . Thus,

$lim inf n → ∞ ⁡ ∫ ℝ N ∫ ℝ N | u n ⁢ ( x ) - u n ⁢ ( y ) | p | x - y | N + p ⁢ s ⁢ 𝑑 y ⁢ η ⁢ ( x ) ⁢ 𝑑 x ≥ ∫ ℝ N ∫ ℝ N | u ⁢ ( x ) - u ⁢ ( y ) | p | x - y | N + p ⁢ s ⁢ 𝑑 y ⁢ η ⁢ ( x ) ⁢ 𝑑 x .$

It follows from $∫ℝN|un⁢(x)-un⁢(y)|p|x-y|N+p⁢s⁢𝑑y⇀μ$ weakly * in $ℳ⁢(ℝN)$ that

$lim n → ∞ ⁡ ∫ ℝ N ∫ ℝ N | u n ⁢ ( x ) - u n ⁢ ( y ) | p | x - y | N + p ⁢ s ⁢ 𝑑 y ⁢ η ⁢ ( x ) ⁢ 𝑑 x = ∫ ℝ N η ⁢ 𝑑 μ .$

Hence,

$∫ ℝ N η ⁢ 𝑑 μ ≥ ∫ ℝ N ∫ ℝ N | u ⁢ ( x ) - u ⁢ ( y ) | p | x - y | N + p ⁢ s ⁢ 𝑑 y ⁢ η ⁢ ( x ) ⁢ 𝑑 x .$

The arbitrariness of $η∈C0⁢(ℝN)$ with $η≥0$ implies that

$μ ≥ ∫ ℝ N | u ⁢ ( x ) - u ⁢ ( y ) | p | x - y | N + p ⁢ s ⁢ 𝑑 y .$

Therefore, we obtain

$μ - ∫ ℝ N | u ⁢ ( x ) - u ⁢ ( y ) | p | x - y | N + p ⁢ s ⁢ 𝑑 y = ∑ j ∈ J μ j ⁢ δ x j + μ ~ .$

Part 3. We show that

$ν = | u | p s * + ∑ j ∈ J ν j ⁢ δ x j ,$

where ${xj}j$ is as above and ${νj}j⊂[0,∞)$ . Since $un⇀u$ weakly in $Ds,p⁢(ℝN)$ , there exists a subsequence still denoted by ${un}n$ such that $un→u$ a.e. in $ℝN$ . Take $η∈C0⁢(ℝN)$ . It follows from the boundedness of ${un}n$ in $Lps*⁢(ℝN)$ and the Brézis–Lieb lemma that

$lim n → ∞ ⁡ ∫ ℝ N ( | u n | p s * - | u n - u | p s * ) ⁢ η ⁢ 𝑑 x = ∫ ℝ N | u | p s * ⁢ η ⁢ 𝑑 x .$

Set $ν¯=ν-|u|ps*$ . Since $∫ℝN|un|ps*⁢η⁢𝑑x→∫ℝNη⁢𝑑ν$ as $n→∞$ , it follows that

(2.1)

$∫ ℝ N η ⁢ 𝑑 ν ¯ = ∫ ℝ N η ⁢ 𝑑 ν - ∫ ℝ N | u | p s * ⁢ η ⁢ 𝑑 x = lim n → ∞ ⁡ ∫ ℝ N | u n - u | p s * ⁢ η ⁢ 𝑑 x ,$

so that $|un-u|ps*⇀ν¯$ weakly * in $ℳ⁢(ℝN)$ . Furthermore,

$ν ¯ = ν - | u | p s * = ∑ j ∈ J ′ ν j ⁢ δ y j + ν ~ .$

Next we prove that the atom of ν is that of μ and $ν~=0$ . Let $x0∈ℝN$ fixed and let $φ∈C0∞⁢(ℝN)$ such that $0≤φ≤1$ , $φ≡1$ in $B⁢(0,1)$ , $φ≡0$ in $ℝN∖B⁢(0,2)$ , and $|∇⁡φ|≤2$ . Denote $φε,0⁢(x)=φ⁢(x-x0ε)$ for all $x∈ℝN$ . Then

$∫ ℝ N | u n ⁢ φ ε , 0 | p s * ⁢ 𝑑 x = ∫ ℝ N | u n | p s * ⁢ φ ε , 0 p s * ⁢ 𝑑 x → ∫ ℝ N φ ε , 0 p s * ⁢ 𝑑 ν as ⁢ n → ∞ ,$

and

$∫ ℝ N φ ε , 0 p s * ⁢ 𝑑 ν → ν ⁢ ( { x 0 } ) as ⁢ ε → 0 .$

Similarly, we have

$∬ ℝ 2 ⁢ N | u n ⁢ ( x ) - u n ⁢ ( y ) | p ⁢ φ ε , 0 p | x - y | N + p ⁢ s ⁢ 𝑑 x ⁢ 𝑑 y → ∫ ℝ N φ ε , 0 p ⁢ 𝑑 μ as ⁢ n → ∞ ,$

and

$∫ ℝ N φ ε , 0 p ⁢ 𝑑 μ → μ ⁢ ( { x 0 } ) as ⁢ ε → 0 .$

Hence, we obtain

(2.2)

$lim ε → 0 ⁡ lim n → ∞ ⁡ ∫ ℝ N | u n ⁢ φ ε , 0 | p s * ⁢ 𝑑 x = ν ⁢ ( { x 0 } )$

and

(2.3)

$lim ε → 0 ⁡ lim n → ∞ ⁡ ∬ ℝ 2 ⁢ N | u n ⁢ ( x ) - u n ⁢ ( y ) | p ⁢ φ ε , 0 p | x - y | N + p ⁢ s ⁢ 𝑑 x ⁢ 𝑑 y = μ ⁢ ( { x 0 } ) .$

Now let us recall the following Young inequality:

(2.4)

$| ζ 1 + ζ 2 | p ≤ ( 1 + β ) p - 1 ⁢ | ζ 1 | p + ( 1 + 1 / β ) p - 1 ⁢ | ζ 2 | p ,$

where $ζ1,ζ2∈ℝ$ and $β>0$ . By the definition of S and inequality (2.4), we get

$∫ ℝ N | u n ⁢ φ ε , 0 | p s * ⁢ 𝑑 x ≤ S - p s * / p ⁢ ( ∬ ℝ 2 ⁢ N | u n ⁢ ( x ) ⁢ φ ε , 0 ⁢ ( x ) - u n ⁢ ( y ) ⁢ φ ε , 0 ⁢ ( y ) | p | x - y | N + p ⁢ s ⁢ 𝑑 x ⁢ 𝑑 y ) p s * / p$

$≤ S - p s * / p ( ( 1 + β ) p - 1 ∬ ℝ 2 ⁢ N | u n ⁢ ( x ) | p ⁢ | φ ε , 0 ⁢ ( x ) - φ ε , 0 ⁢ ( y ) | p | x - y | N + p ⁢ s d x d y$

(2.5)

$+ ( 1 + 1 / β ) p - 1 ∬ ℝ 2 ⁢ N | u n ⁢ ( x ) - u n ⁢ ( y ) | p ⁢ | φ ε , 0 ⁢ ( y ) | p | x - y | N + p ⁢ s d x d y ) p s * / p$

for any $β>0$ . By (2.2), (2.3), (2.5), and Lemma 2.3, we deduce

$ν ⁢ ( { x 0 } ) ≤ ( 1 + 1 / β ) p s * ⁢ ( p - 1 ) / p ⁢ S - p s * / p ⁢ μ ⁢ ( { x 0 } ) p s * / p .$

Then letting $β→∞$ , we obtain

(2.6)

$ν ⁢ ( { x 0 } ) ≤ S - p s * / p ⁢ μ ⁢ ( { x 0 } ) p s * / p .$

Hence, the arbitrariness of $x0$ implies that the atom of ν is that of μ, that is ${yj:j∈J′}⊂{xj:j∈J}$ . Therefore, we get

$ν - | u | p s * = ∑ j ∈ J ν j ⁢ δ x j + ν ~ .$

It remains to show that $ν~=0$ . To this end, let $u¯n=un-u$ . Then $u¯n⇀0$ weakly in $Ds,p⁢(ℝN)$ . Hence there exists a subsequence of ${u¯n}n$ still denoted by ${u¯n}n$ such that

$∫ ℝ N | u ¯ n ⁢ ( x ) - u ¯ n ⁢ ( y ) | p | x - y | N + p ⁢ s ⁢ 𝑑 y ⇀ μ ¯ weakly in ⁢ ℳ ⁢ ( ℝ N ) .$

For any $0<r<R$ take $η∈C0∞⁢(BR⁢(x0))$ satisfying $0≤η≤1$ and $η≡1$ on $Br⁢(x0)$ . It follows from the definition of S that

$∫ B R ⁢ ( x 0 ) η p s * ⁢ | u ¯ n | p s * ⁢ 𝑑 x = ∫ B R ⁢ ( x 0 ) | η ⁢ u ¯ n | p s * ⁢ 𝑑 x ≤ S - p s * / p ⁢ ( ∫ B R ⁢ ( x 0 ) ∫ B R ⁢ ( x 0 ) | η ⁢ ( x ) ⁢ u ¯ n ⁢ ( x ) - η ⁢ ( y ) ⁢ u ¯ n ⁢ ( y ) | p | x - y | N + p ⁢ s ⁢ 𝑑 y ⁢ 𝑑 x ) p s * / p$

$≤ S - p s * / p ( ∫ B R ⁢ ( x 0 ) ∫ B R ⁢ ( x 0 ) 2 p - 1 ⁢ | ( η ⁢ ( x ) - η ⁢ ( y ) ) ⁢ u ¯ n ⁢ ( y ) | p | x - y | N + p ⁢ s d y d x$

(2.7)

$+ ∫ B R ⁢ ( x 0 ) ∫ B R ⁢ ( x 0 ) 2 p - 1 ⁢ | η ⁢ ( x ) ⁢ ( u ¯ n ⁢ ( x ) - u ¯ n ⁢ ( y ) ) | p | x - y | N + p ⁢ s d y d x ) p s * / p .$

Note that

$| η ⁢ ( x ) - η ⁢ ( y ) | p ≤ ( ∥ η ∥ C 1 + 2 ) p ⁢ min ⁡ { 1 , | x - y | p }$

for all $x,y∈BR⁢(x0)$ . Hence, by the compact embedding for fractional Sobolev spaces on bounded domains, we obtain that $u¯n→0$ strongly in $Lp⁢(BR⁢(x0))$ . Furthermore, we deduce

$∫ B R ⁢ ( x 0 ) ∫ B R ⁢ ( x 0 ) | ( η ⁢ ( x ) - η ⁢ ( y ) ) ⁢ u ¯ n ⁢ ( y ) | p | x - y | N + p ⁢ s ⁢ 𝑑 y ⁢ 𝑑 x ≤ ( ∥ η ∥ C 1 + 2 ) p ⁢ ∫ B R ⁢ ( x 0 ) min ⁡ { 1 , | x - y | p } ⁢ | x - y | - N - p ⁢ s ⁢ 𝑑 x ⁢ ∫ B R ⁢ ( x 0 ) | u ¯ n ⁢ ( y ) | p ⁢ 𝑑 y$

$≤ C ⁢ ∫ B R ⁢ ( x 0 ) | u ¯ n ⁢ ( y ) | p ⁢ 𝑑 y → 0 as ⁢ n → ∞ ,$

so that

(2.8)

$lim n → ∞ ⁡ ∫ B R ⁢ ( x 0 ) ∫ B R ⁢ ( x 0 ) | ( η ⁢ ( x ) - η ⁢ ( y ) ) ⁢ u ¯ n ⁢ ( y ) | p | x - y | N + p ⁢ s ⁢ 𝑑 y ⁢ 𝑑 x = 0 .$

Note that

(2.9)

$lim sup n → ∞ ⁡ ∫ B R ⁢ ( x 0 ) ∫ B R ⁢ ( x 0 ) η p ⁢ ( x ) ⁢ | u ¯ n ⁢ ( x ) - u ¯ n ⁢ ( y ) | p | x - y | N + p ⁢ s ⁢ 𝑑 y ⁢ 𝑑 x ≤ ∫ ℝ N η p ⁢ 𝑑 μ ¯ ≤ ∫ B R ⁢ ( x 0 ) ¯ 𝑑 μ ¯ = μ ¯ ⁢ ( B R ⁢ ( x 0 ) ¯ ) .$

Inserting (2.8) and (2.9) into (2.7), we obtain

$ν ¯ ⁢ ( B r ⁢ ( x 0 ) ¯ ) ≤ ∫ ℝ N η p ⁢ 𝑑 ν ¯ = lim n → ∞ ⁡ ∫ ℝ N | u ¯ n | p s * ⁢ η p ⁢ 𝑑 x ≤ ( C * ) - p s * / p ⁢ ( 2 p - 1 ⁢ μ ¯ ⁢ ( B R ⁢ ( x 0 ) ¯ ) ) p s * / p .$

Letting $r→R-$ , we get

(2.10)

$ν ¯ ⁢ ( B R ⁢ ( x 0 ) ¯ ) ≤ S - p s * / p ⁢ ( 2 p - 1 ⁢ μ ¯ ⁢ ( B R ⁢ ( x 0 ) ¯ ) ) p s * / p .$

This means that $ν¯$ is absolutely continuous with respect to $μ¯$ . Hence the Radon–Nikodym theorem implies that there exists a function $h∈L1⁢(ℝN,μ¯)$ such that $d⁢ν¯=h⁢d⁢μ¯$ . Then we derive from Lebesgue’s differential theorem and (2.10) that

$h ⁢ ( x 0 ) = lim R → 0 ⁡ ν ¯ ⁢ ( B R ⁢ ( x 0 ) ¯ ) μ ¯ ⁢ ( B R ⁢ ( x 0 ) ¯ ) ≤ ( C * ) - p s * / p ⁢ 2 p s * ⁢ ( p - 1 ) / p ⁢ lim R → 0 ⁡ μ ¯ ⁢ ( B R ⁢ ( x 0 ) ¯ ) p s * / p - 1$

(2.11)

$= S - p s * / p ⁢ 2 p s * ⁢ ( p - 1 ) / p ⁢ μ ¯ ⁢ ( { x 0 } ) p s * / p - 1 .$

Now we show that $ν~=0$ . Let $x∈ℝN∖{xj:i∈J}$ . If $h⁢(x)≠0$ , then by (2.11) we know that $μ¯⁢({x})≠0$ , thus $ν¯⁢({x})≠0$ . Note that (2.1) implies that $ν¯$ and ν have the same atoms, so that x is an atom of μ, which is a contradiction. Hence $h≡0$ on $ℝN∖{xj:j∈J}$ . Furthermore, by $d⁢ν¯=h⁢d⁢μ¯$ , we obtain $ν¯=0$ on $ℝN∖{xj:j∈J}$ . In conclusion, $ν~=0$ , since $ν~$ is a non-atomic measure.

Part 4. Finally, we prove that

$ν ⁢ ( ℝ N ) ≤ 2 p s * ⁢ ( p - 1 ) / p ⁢ S - p s * / p ⁢ μ ⁢ ( ℝ N ) p s * / p and ν j ≤ S - p s * / p ⁢ μ j p s * / p for each ⁢ j ∈ J .$

Take $η∈C0∞⁢(B2⁢R⁢(0))$ satisfying $0≤η≤1$ , $η≡1$ on $BR⁢(0)$ and $|∇⁡η|≤2/R$ . Let $β>0$ . Then a similar discussion as in Part 3 gives that

$∫ ℝ N η p s * | u n | p s * d x ≤ S - p s * / p ( ( 1 + β ) p - 1 ∫ ℝ N ∫ ℝ N | ( η ⁢ ( x ) - η ⁢ ( y ) ) ⁢ u n ⁢ ( x ) | p | x - y | N + p ⁢ s d y d x$

(2.12)

$+ ( 1 + 1 β ) p - 1 ∫ ℝ N ∫ ℝ N | η ⁢ ( x ) ⁢ ( u n ⁢ ( x ) - u n ⁢ ( y ) ) | p | x - y | N + p ⁢ s d y d x ) p s * / p .$

With a similar discussion as in the proof of Lemma 2.3, we have

$lim R → ∞ ⁡ lim sup n → ∞ ⁡ ∬ ℝ 2 ⁢ N | ( η ⁢ ( x ) - η ⁢ ( y ) ) ⁢ u n ⁢ ( x ) | p | x - y | N + p ⁢ s ⁢ 𝑑 y ⁢ 𝑑 x = 0 .$

Letting $n→∞$ in (2.12), we have

$∫ ℝ N η p s * ⁢ 𝑑 ν ≤ S - p s * / p ⁢ ( ( 1 + 1 β ) p - 1 ⁢ ∫ ℝ N η p ⁢ 𝑑 μ ) p s * / p .$

Letting $β→∞$ , we obtain

(2.13)

$∫ ℝ N η p s * ⁢ 𝑑 ν ≤ S - p s * / p ⁢ ( ∫ ℝ N η p ⁢ 𝑑 μ ) p s * / p .$

Combining (2.13) with $ν⁢(BR⁢(0)¯)≤∫ℝNηps*⁢𝑑ν$ , we get

$ν ⁢ ( ℝ N ) ≤ S - p s * / p ⁢ ( μ ⁢ ( ℝ N ) ) p s * / p .$

With a discussion similar to (2.6), we can conclude that

$ν j ≤ S - p s * / p ⁢ μ j p s * / p for each ⁢ j ∈ J .$

Indeed, this fact follows by replacing $φε,0$ in Part 3 with $φε,j=φ⁢((x-xj)/ε)$ . Thus, the proof is complete. ∎

Actually, Theorem 2.2 does not provide any information about the possible loss of mass at infinity for a weakly convergent sequence. The following theorem expresses this fact in quantitative terms.

Theorem 2.4.

Let ${un}n⊂Ds,p⁢(RN)$ be a bounded sequence such that

${ ∫ ℝ N | u n ⁢ ( x ) - u n ⁢ ( y ) | p | x - y | N + p ⁢ s ⁢ 𝑑 y ⇀ μ weakly * in ⁢ ℳ ⁢ ( ℝ N ) , | u n | p s * ⇀ ν weakly * in ⁢ ℳ ⁢ ( ℝ N ) ,$

and define

$μ ∞ = lim R → ∞ ⁡ lim sup n → ∞ ⁡ ∫ { x ∈ ℝ N : | x | > R } ∫ ℝ N | u n ⁢ ( x ) - u n ⁢ ( y ) | p | x - y | N + p ⁢ s ⁢ 𝑑 y ⁢ 𝑑 x$

and

$ν ∞ = lim R → ∞ ⁡ lim sup n → ∞ ⁡ ∫ { x ∈ ℝ N : | x | > R } | u n | p s * ⁢ 𝑑 x .$

Then the quantities $μ∞$ and $ν∞$ are well defined and satisfy

$lim sup n → ∞ ⁡ ∫ ℝ N ∫ ℝ N | u n ⁢ ( x ) - u n ⁢ ( y ) | p | x - y | N + p ⁢ s ⁢ 𝑑 y ⁢ 𝑑 x = ∫ ℝ N 𝑑 μ + μ ∞$

and

$lim sup n → ∞ ⁡ ∫ ℝ N | u n | p s * ⁢ 𝑑 x = ∫ ℝ N 𝑑 ν + ν ∞ .$

Moreover, the following inequality holds:

$S ⁢ ν ∞ p / p s * ≤ μ ∞ .$

Proof.

Let $χ∈C∞⁢(ℝN)$ such that $0≤χ≤1$ , $χ=1$ in $ℝN∖B2⁢(0)$ , $χ≡0$ in $B1⁢(0)$ , and $|∇⁡χ|≤2$ . For any $R>0$ define $χR⁢(x)=χ⁢(x/R)$ . Then

$∫ { x ∈ ℝ N : | x | > 2 ⁢ R } ∫ ℝ N | u n ⁢ ( x ) - u n ⁢ ( y ) | p | x - y | N + p ⁢ s ⁢ 𝑑 y ⁢ 𝑑 x ≤ ∫ ℝ N ( ∫ ℝ N | u n ⁢ ( x ) - u n ⁢ ( y ) | p | x - y | N + p ⁢ s ⁢ 𝑑 y ) ⁢ χ R p ⁢ 𝑑 x$

$≤ ∫ { x ∈ ℝ N : | x | > R } ( ∫ ℝ N | u n ⁢ ( x ) - u n ⁢ ( y ) | p | x - y | N + p ⁢ s ⁢ 𝑑 y ) ⁢ 𝑑 x .$

This means that

$μ ∞ = lim R → ∞ ⁡ lim sup n → ∞ ⁡ ∫ ℝ N ( ∫ ℝ N | u n ⁢ ( x ) - u n ⁢ ( y ) | p | x - y | N + p ⁢ s ⁢ 𝑑 y ) ⁢ χ R p ⁢ 𝑑 x .$

A similar discussion gives that

$ν ∞ = lim R → ∞ ⁡ lim sup n → ∞ ⁡ ∫ ℝ N | u n | p s * ⁢ η R ⁢ 𝑑 x = lim R → ∞ ⁡ lim sup n → ∞ ⁡ ∫ ℝ N | u n ⁢ χ R | p s * ⁢ 𝑑 x .$

Note that

$∫ ℝ N ( ∫ ℝ N | u n ⁢ ( x ) - u n ⁢ ( y ) | p | x - y | N + p ⁢ s ⁢ 𝑑 y ) ⁢ 𝑑 x$

(2.14)

$= ∫ ℝ N ( ∫ ℝ N | u n ⁢ ( x ) - u n ⁢ ( y ) | p | x - y | N + p ⁢ s ⁢ 𝑑 y ) ⁢ χ R p ⁢ 𝑑 x + ∫ ℝ N ( ∫ ℝ N | u n ⁢ ( x ) - u n ⁢ ( y ) | p | x - y | N + p ⁢ s ⁢ 𝑑 y ) ⁢ ( 1 - χ R p ) ⁢ 𝑑 x .$

It is easy to see that

$∫ ℝ N ( ∫ ℝ N | u n ⁢ ( x ) - u n ⁢ ( y ) | p | x - y | N + p ⁢ s ⁢ 𝑑 y ) ⁢ ( 1 - χ R p ) ⁢ 𝑑 x → ∫ ℝ N ( 1 - χ R p ) ⁢ 𝑑 μ ,$

as $n→∞$ . Hence, we get

$μ ⁢ ( ℝ N ) = lim R → ∞ ⁡ lim n → ∞ ⁡ ∫ ℝ N ( ∫ ℝ N | u n ⁢ ( x ) - u n ⁢ ( y ) | p | x - y | N + p ⁢ s ⁢ 𝑑 y ) ⁢ ( 1 - χ R p ) ⁢ 𝑑 x .$

It follows from (2.14) that

$lim sup n → ∞ ⁡ ∫ ℝ N ( ∫ ℝ N | u n ⁢ ( x ) - u n ⁢ ( y ) | p | x - y | N + p ⁢ s ⁢ 𝑑 y ) ⁢ 𝑑 x = lim R → ∞ ⁡ lim sup n → ∞ ⁡ ∫ ℝ N ( ∫ ℝ N | u n ⁢ ( x ) - u n ⁢ ( y ) | p | x - y | N + p ⁢ s ⁢ 𝑑 y ) ⁢ 𝑑 x$

$= lim R → ∞ ⁡ lim sup n → ∞ ⁡ [ ∫ ℝ N ( ∫ ℝ N | u n ⁢ ( x ) - u n ⁢ ( y ) | p | x - y | N + p ⁢ s ⁢ 𝑑 y ) ⁢ χ R p ⁢ 𝑑 x + ∫ ℝ N ( 1 - χ R p ) ⁢ 𝑑 μ ]$

$= lim R → ∞ ⁡ lim sup n → ∞ ⁡ ∫ ℝ N ( ∫ ℝ N | u n ⁢ ( x ) - u n ⁢ ( y ) | p | x - y | N + p ⁢ s ⁢ 𝑑 y ) ⁢ χ R p ⁢ 𝑑 x + μ ⁢ ( ℝ N )$

$= μ ∞ + μ ⁢ ( ℝ N ) .$

Similarly, we can obtain that

$lim sup n → ∞ ⁡ ∫ ℝ N | u n | p s * ⁢ 𝑑 x = ν ⁢ ( ℝ N ) + ν ∞ .$

Finally, we prove that $S⁢ν∞p/ps*≤μ∞$ . For this purpose, we first show that

(2.15)

$lim R → ∞ ⁡ lim sup n → ∞ ⁡ ∬ ℝ 2 ⁢ N | u n | p ⁢ | χ R ⁢ ( x ) - χ R ⁢ ( y ) | p | x - y | N + p ⁢ s ⁢ 𝑑 x ⁢ 𝑑 y = 0 .$

Similar to Lemma 2.3, we know that

$ℝ N × ℝ N = ( ( ℝ N ∖ B 2 ⁢ R ⁢ ( 0 ) ) × ( ℝ N ∖ B 2 ⁢ R ⁢ ( 0 ) ) ) ∪ ( ( ℝ N ∖ B 2 ⁢ R ⁢ ( 0 ) ) × B 2 ⁢ R ⁢ ( 0 ) ) ∪ ( B 2 ⁢ R ⁢ ( 0 ) × ℝ N ) .$

Thus, we divide the discussion into three steps.

Step 1. For all $(x,y)∈(ℝN∖B2⁢R⁢(0))×(ℝN∖B2⁢R⁢(0))$ it is easy to see that $χR⁢(x)=χR⁢(y)=1$ . Thus,

$∫ ℝ N ∖ B 2 ⁢ R ⁢ ( 0 ) ∫ ℝ N ∖ B 2 ⁢ R ⁢ ( 0 ) | u n | p ⁢ | χ R ⁢ ( x ) - χ R ⁢ ( y ) | p | x - y | N + p ⁢ s ⁢ 𝑑 x ⁢ 𝑑 y = 0 .$

Step 2. Let $k>4$ . Clearly, we have

$( ℝ N ∖ B 2 ⁢ R ⁢ ( 0 ) ) × B 2 ⁢ R ⁢ ( 0 ) = ( ( ℝ 2 ⁢ N ∖ B k ⁢ R ⁢ ( 0 ) ) × B 2 ⁢ R ⁢ ( 0 ) ) ∪ ( ( B k ⁢ R ⁢ ( 0 ) ∖ B 2 ⁢ R ⁢ ( 0 ) ) × B 2 ⁢ R ⁢ ( 0 ) ) .$

If $(x,y)∈ℝ2⁢N∖Bk⁢R⁢(0)×B2⁢R⁢(0)$ , then

$| x - y | ≥ | x | - | y | ≥ | x | - 2 ⁢ R > | x | 2 .$

Hence,

$∫ ℝ N ∖ B k ⁢ R ⁢ ( 0 ) ∫ B 2 ⁢ R ⁢ ( 0 ) | u n ⁢ ( x ) | p ⁢ | χ R ⁢ ( x ) - χ R ⁢ ( y ) | p | x - y | N + p ⁢ s ⁢ 𝑑 x ⁢ 𝑑 y ≤ 2 p + N + p ⁢ s ⁢ ∫ ℝ N ∖ B k ⁢ R ⁢ ( 0 ) ∫ B 2 ⁢ R ⁢ ( 0 ) | u n ⁢ ( x ) | p | x | N + p ⁢ s ⁢ 𝑑 x ⁢ 𝑑 y$

$≤ C ⁢ R N ⁢ ∫ ℝ N ∖ B k ⁢ R ⁢ ( 0 ) | u n ⁢ ( x ) | p | x | N + p ⁢ s ⁢ 𝑑 x$

$≤ C ⁢ R N ⁢ ( ∫ ℝ N ∖ B k ⁢ R ⁢ ( 0 ) | u n ⁢ ( x ) | p s * ⁢ 𝑑 x ) p / p s * ⁢ ( ∫ ℝ N ∖ B k ⁢ R ⁢ ( 0 ) 1 | x | N 2 s ⁢ p + N ⁢ 𝑑 x ) s ⁢ p / N$

$≤ C k N ⁢ ( ∫ ℝ N ∖ B k ⁢ R ⁢ ( 0 ) | u n ⁢ ( x ) | p s * ⁢ 𝑑 x ) p / p s * ≤ C k N .$

Note that

$∫ B k ⁢ R ⁢ ( 0 ) ∖ B 2 ⁢ R ⁢ ( 0 ) ∫ B 2 ⁢ R ⁢ ( 0 ) | u n ⁢ ( x ) | p ⁢ | χ R ⁢ ( x ) - χ R ⁢ ( y ) | p | x - y | N + p ⁢ s ⁢ 𝑑 x ⁢ 𝑑 y ≤ C R p ⁢ ∫ B k ⁢ R ⁢ ( 0 ) ∖ B 2 ⁢ R ⁢ ( 0 ) ∫ B 2 ⁢ R ⁢ ( 0 ) | u n ⁢ ( x ) | p | x - y | N + ( s - 1 ) ⁢ p ⁢ 𝑑 x ⁢ 𝑑 y$

$≤ C R p ⁢ ( k ⁢ R ) ( 1 - s ) ⁢ p ⁢ ∫ B k ⁢ R ⁢ ( 0 ) ∖ B 2 ⁢ R ⁢ ( 0 ) | u n ⁢ ( x ) | p ⁢ 𝑑 x$

$= C ⁢ k ( 1 - s ) ⁢ p R s ⁢ p ⁢ ∫ B k ⁢ R ⁢ ( 0 ) ∖ B 2 ⁢ R ⁢ ( 0 ) | u n ⁢ ( x ) | p ⁢ 𝑑 x .$

Therefore, we obtain

$∫ ℝ N ∖ B 2 ⁢ R ⁢ ( 0 ) ∫ B 2 ⁢ R ⁢ ( 0 ) | u n ⁢ ( x ) | p ⁢ | χ R ⁢ ( x ) - χ R ⁢ ( y ) | p | x - y | N + p ⁢ s ⁢ 𝑑 x ⁢ 𝑑 y ≤ C k N + C ⁢ k ( 1 - s ) ⁢ p R s ⁢ p ⁢ ∫ B k ⁢ R ⁢ ( 0 ) ∖ B 2 ⁢ R ⁢ ( 0 ) | u n ⁢ ( x ) | p ⁢ 𝑑 x .$

Step 3. Let $ε∈(0,1)$ fixed. Then

$∫ B 2 ⁢ R ⁢ ( 0 ) ∫ ℝ N | u n ⁢ ( x ) | p ⁢ | χ R ⁢ ( x ) - χ R ⁢ ( y ) | p | x - y | N + p ⁢ s ⁢ 𝑑 x ⁢ 𝑑 y$

$≤ ∫ B 2 ⁢ R ⁢ ( 0 ) ∖ B ε ⁢ R ⁢ ( 0 ) ∫ ℝ N | u n ⁢ ( x ) | p ⁢ | χ R ⁢ ( x ) - χ R ⁢ ( y ) | p | x - y | N + p ⁢ s ⁢ 𝑑 x ⁢ 𝑑 y + ∫ B ε ⁢ R ⁢ ( 0 ) ∫ ℝ N | u n ⁢ ( x ) | p ⁢ | χ R ⁢ ( x ) - χ R ⁢ ( y ) | p | x - y | N + p ⁢ s ⁢ 𝑑 x ⁢ 𝑑 y .$

Note that

$∫ B 2 ⁢ R ⁢ ( 0 ) ∖ B ε ⁢ R ⁢ ( 0 ) ∫ ℝ N ∩ { y : | x - y | < R } | u n ⁢ ( x ) | p ⁢ | χ R ⁢ ( x ) - χ R ⁢ ( y ) | p | x - y | N + p ⁢ s ⁢ 𝑑 x ⁢ 𝑑 y ≤ C R s ⁢ p ⁢ ∫ B 2 ⁢ R ⁢ ( 0 ) ∖ B ε ⁢ R ⁢ ( 0 ) | u n ⁢ ( x ) | p ⁢ 𝑑 x$

and

$∫ B 2 ⁢ R ⁢ ( 0 ) ∖ B ε ⁢ R ⁢ ( 0 ) ∫ ℝ N ∩ { y : | x - y | ≥ R } | u n ⁢ ( x ) | p ⁢ | χ R ⁢ ( x ) - χ R ⁢ ( y ) | p | x - y | N + p ⁢ s ⁢ 𝑑 x ⁢ 𝑑 y ≤ C R s ⁢ p ⁢ ∫ B 2 ⁢ R ⁢ ( 0 ) ∖ B ε ⁢ R ⁢ ( 0 ) | u n ⁢ ( x ) | p ⁢ 𝑑 x .$

Hence,

$∫ B 2 ⁢ R ⁢ ( 0 ) ∖ B ε ⁢ R ⁢ ( 0 ) ∫ ℝ N | u n ⁢ ( x ) | p ⁢ | χ R ⁢ ( x ) - χ R ⁢ ( y ) | p | x - y | N + p ⁢ s ⁢ 𝑑 x ⁢ 𝑑 y ≤ C R s ⁢ p ⁢ ∫ B 2 ⁢ R ⁢ ( 0 ) ∖ B ε ⁢ R ⁢ ( 0 ) | u n ⁢ ( x ) | p ⁢ 𝑑 x .$

By the definition of $χR$ and the choice of ε, it follows easily that

$∫ B ε ⁢ R ⁢ ( 0 ) ∫ ℝ N | u n ⁢ ( x ) | p ⁢ | χ R ⁢ ( x ) - χ R ⁢ ( y ) | p | x - y | N + p ⁢ s ⁢ 𝑑 x ⁢ 𝑑 y = ∫ B ε ⁢ R ⁢ ( 0 ) ∫ ℝ N ∖ B R ⁢ ( 0 ) | u n ⁢ ( x ) | p ⁢ | χ R ⁢ ( x ) - χ R ⁢ ( y ) | p | x - y | N + p ⁢ s ⁢ 𝑑 x ⁢ 𝑑 y .$

If $(x,y)∈Bε⁢R⁢(0)×(ℝN∖BR⁢(0))$ , then $|x-y|>(1-ε)⁢R$ . Thus,

$∫ B ε ⁢ R ⁢ ( 0 ) ∫ ℝ N ∖ B R ⁢ ( 0 ) | u n ⁢ ( x ) | p ⁢ | χ R ⁢ ( x ) - χ R ⁢ ( y ) | p | x - y | N + p ⁢ s ⁢ 𝑑 x ⁢ 𝑑 y ≤ 2 p ⁢ ∫ B ε ⁢ R ⁢ ( 0 ) ∫ ℝ N ∖ B R ⁢ ( 0 ) | u n ⁢ ( x ) | p | x - y | N + p ⁢ s ⁢ 𝑑 x ⁢ 𝑑 y$

$≤ C ⁢ ∫ B ε ⁢ R ⁢ ( 0 ) | u n | p ⁢ 𝑑 x ⁢ ∫ ( 1 - ε ) ⁢ R ∞ 1 r 1 + s ⁢ p ⁢ 𝑑 r$

$= C [ ( 1 - ε ) ⁢ R ] s ⁢ p ⁢ ∫ B ε ⁢ R ⁢ ( 0 ) | u n | p ⁢ 𝑑 x .$

Thus, we get

$∫ B 2 ⁢ R ⁢ ( 0 ) ∫ ℝ N | u n ⁢ ( x ) | p ⁢ | χ R ⁢ ( x ) - χ R ⁢ ( y ) | p | x - y | N + p ⁢ s ⁢ 𝑑 x ⁢ 𝑑 y ≤ C R s ⁢ p ⁢ ∫ B 2 ⁢ R ⁢ ( 0 ) ∖ B ε ⁢ R ⁢ ( 0 ) | u n ⁢ ( x ) | p ⁢ 𝑑 x + C [ ( 1 - ε ) ⁢ R ] s ⁢ p ⁢ ∫ B ε ⁢ R ⁢ ( 0 ) | u n | p ⁢ 𝑑 x .$

From Steps 1–3 we deduce

$∬ ℝ 2 ⁢ N | u n ⁢ ( x ) | p ⁢ | χ R ⁢ ( x ) - χ R ⁢ ( y ) | p | x - y | N + p ⁢ s ⁢ 𝑑 x ⁢ 𝑑 y ≤ C k N + C ⁢ k ( 1 - s ) ⁢ p R s ⁢ p ⁢ ∫ B k ⁢ R ⁢ ( 0 ) ∖ B 2 ⁢ R ⁢ ( 0 ) | u n ⁢ ( x ) | p ⁢ 𝑑 x$

$+ C R s ⁢ p ⁢ ∫ B 2 ⁢ R ⁢ ( 0 ) ∖ B ε ⁢ R ⁢ ( 0 ) | u n ⁢ ( x ) | p ⁢ 𝑑 x + C [ ( 1 - ε ) ⁢ R ] s ⁢ p ⁢ ∫ B ε ⁢ R ⁢ ( 0 ) | u n | p ⁢ 𝑑 x .$

Since ${un}n$ is bounded in $Ds,p⁢(ℝN)$ and the embedding $Ds,p⁢(ℝN)↪Llocp⁢(ℝN)$ is compact, without loss of generality, we assume that $un→u$ in $Llocp⁢(ℝN)$ for some $u∈Ds,p⁢(ℝN)$ . Then we have

$∬ ℝ 2 ⁢ N | u n ⁢ ( x ) | p ⁢ | χ R ⁢ ( x ) - χ R ⁢ ( y ) | p | x - y | N + p ⁢ s ⁢ 𝑑 x ⁢ 𝑑 y$

$≤ C k N + C ⁢ k ( 1 - s ) ⁢ p R s ⁢ p ⁢ ∫ B k ⁢ R ⁢ ( 0 ) ∖ B 2 ⁢ R ⁢ ( 0 ) | u n ⁢ ( x ) | p ⁢ 𝑑 x + C R s ⁢ p ⁢ ∫ B 2 ⁢ R ⁢ ( 0 ) ∖ B ε ⁢ R ⁢ ( 0 ) | u n ⁢ ( x ) | p ⁢ 𝑑 x + C [ ( 1 - ε ) ⁢ R ] s ⁢ p ⁢ ∫ B ε ⁢ R ⁢ ( 0 ) | u n | p ⁢ 𝑑 x$

$→ C k N + C ⁢ k ( 1 - s ) ⁢ p R s ⁢ p ⁢ ∫ B k ⁢ R ⁢ ( 0 ) ∖ B 2 ⁢ R ⁢ ( 0 ) | u ⁢ ( x ) | p ⁢ 𝑑 x + C R s ⁢ p ⁢ ∫ B 2 ⁢ R ⁢ ( 0 ) ∖ B ε ⁢ R ⁢ ( 0 ) | u ⁢ ( x ) | p ⁢ 𝑑 x + C [ ( 1 - ε ) ⁢ R ] s ⁢ p ⁢ ∫ B ε ⁢ R ⁢ ( 0 ) | u | p ⁢ 𝑑 x$

as $n→∞$ . In view of the Hölder inequality, we get

$lim sup n → ∞ ⁡ ∬ ℝ 2 ⁢ N | u n ⁢ ( x ) | p ⁢ | χ R ⁢ ( x ) - χ R ⁢ ( y ) | p | x - y | N + p ⁢ s ⁢ 𝑑 x ⁢ 𝑑 y$

$≤ C k N + C ⁢ k ( 1 - s ) ⁢ p R s ⁢ p ⁢ ∫ B k ⁢ R ⁢ ( 0 ) ∖ B 2 ⁢ R ⁢ ( 0 ) | u ⁢ ( x ) | p ⁢ 𝑑 x + C R s ⁢ p ⁢ ∫ B 2 ⁢ R ⁢ ( 0 ) ∖ B ε ⁢ R ⁢ ( 0 ) | u ⁢ ( x ) | p ⁢ 𝑑 x + C [ ( 1 - ε ) ⁢ R ] s ⁢ p ⁢ ∫ B ε ⁢ R ⁢ ( 0 ) | u | p ⁢ 𝑑 x$

$≤ C k N + C ⁢ k p ⁢ ( ∫ B k ⁢ R ⁢ ( 0 ) ∖ B 2 ⁢ R ⁢ ( 0 ) | u ⁢ ( x ) | p s * ⁢ 𝑑 x ) p / p s * + C ⁢ ( ∫ B 2 ⁢ R ⁢ ( 0 ) ∖ B ε ⁢ R ⁢ ( 0 ) | u ⁢ ( x ) | p s * ⁢ 𝑑 x ) p / p s *$

$+ C ⁢ ( ε 1 - ε ) s ⁢ p ⁢ ( ∫ B ε ⁢ R ⁢ ( 0 ) | u ⁢ ( x ) | p s * ⁢ 𝑑 x ) p / p s * .$

Since $∫ℝN|u|ps*⁢𝑑x<∞$ , we obtain

$lim R → ∞ ⁡ ∫ B k ⁢ R ⁢ ( 0 ) ∖ B 2 ⁢ R ⁢ ( 0 ) | u ⁢ ( x ) | p s * ⁢ 𝑑 x = lim R → ∞ ⁡ ∫ B 2 ⁢ R ⁢ ( 0 ) ∖ B ε ⁢ R ⁢ ( 0 ) | u ⁢ ( x ) | p s * ⁢ 𝑑 x = 0$

for $k>4$ and $ε∈(0,1)$ . Now we take $ε=1/k$ . Then

$lim R → ∞ ⁡ lim sup n → ∞ ⁡ ∬ ℝ 2 ⁢ N | u n ⁢ ( x ) | p ⁢ | χ R ⁢ ( x ) - χ R ⁢ ( y ) | p | x - y | N + p ⁢ s ⁢ 𝑑 x ⁢ 𝑑 y ≤ C k N + C ⁢ ( 1 k - 1 ) s ⁢ p ⁢ ( ∫ ℝ N | u ⁢ ( x ) | p s * ⁢ 𝑑 x ) p / p s * → 0$

as $k→∞$ . Hence, we get the desired result (2.15).

Let $β>0$ . By (2.4), we have

$∫ ℝ N χ R p s * | u n | p s * d x ≤ S - p s * / p ( ( 1 + β ) p - 1 ∫ ℝ N ∫ ℝ N | ( χ R ⁢ ( x ) - χ R ⁢ ( y ) ) ⁢ u n ⁢ ( x ) | p | x - y | N + p ⁢ s d y d x$

$+ ( 1 + 1 β ) p - 1 ∫ ℝ N ∫ ℝ N | χ R ⁢ ( x ) ⁢ ( u n ⁢ ( x ) - u n ⁢ ( y ) ) | p | x - y | N + p ⁢ s d y d x ) p s * / p .$

By (2.15), we have

$ν ∞ = lim sup n → ∞ ⁡ ∫ ℝ N χ R p s * ⁢ | u n | p s * ⁢ 𝑑 x$

$≤ S - p s * / p ⁢ ( ( 1 + 1 β ) p - 1 ⁢ lim sup n → ∞ ⁡ ∫ ℝ N ∫ ℝ N | χ R ⁢ ( x ) ⁢ ( u n ⁢ ( x ) - u n ⁢ ( y ) ) | p | x - y | N + p ⁢ s ⁢ 𝑑 y ⁢ 𝑑 x ) p s * / p$

$≤ S - p s * / p ⁢ ( 1 + 1 β ) ( p - 1 ) ⁢ p / p s * ⁢ μ ∞ p s * / p .$

Letting $β→∞$ , we get $ν∞≤S-ps*/p⁢μ∞ps*/p$ . Therefore, the proof is complete. ∎

3 Proofs of Theorems 1.2 and 1.3

Inspired by the idea in [7] and [26], using the solutions $Uε$ of problem (1.3), we give the proofs of Theorems 1.2 and 1.3 directly.

Proof of Theorem 1.2.

For any $μ>0$ and $Uε$ in (1.4) define $vε,μ=μ1/(ps*-p)⁢Uε$ . Then $vε,μ$ weakly solves the following equation:

$μ ⁢ ( - Δ ) p s ⁢ v ε , μ = | v ε , μ | p s * - 2 ⁢ v ε , μ .$

Now we consider the following equation:

(3.1)

$μ = a + b ⁢ ( ∬ ℝ 2 ⁢ N | v ε , μ ⁢ ( x ) - v ε , μ ⁢ ( y ) | p | x - y | N + p ⁢ s ⁢ 𝑑 x ⁢ 𝑑 y ) θ - 1 = a + b ⁢ S N ⁢ ( θ - 1 ) / ( p ⁢ s ) ⁢ μ p ⁢ ( θ - 1 ) / ( p s * - p ) .$

(i) For $θ=N/(N-s⁢p)$ , $a>0$ and $b<S-N⁢(θ-1)/(p⁢s)=S-θ$ we have that $μ0=a/(1-b⁢Sθ)$ is a solution of equation (3.1). For $θ=(N+s⁢p)/(N-s⁢p)$ and $a,b>0$ satisfying $4⁢a⁢b⁢Sθ+1<1$ we have that

$μ 1 = 1 - 1 - 4 ⁢ a ⁢ b ⁢ S θ + 1 2 ⁢ b ⁢ S θ + 1 and μ 2 = 1 + 1 - 4 ⁢ a ⁢ b ⁢ S θ + 1 2 ⁢ b ⁢ S θ + 1$

are solutions of (3.1). Hence, $vε,μi=μi1/(ps*-p)⁢Uε$ ( $i=0,1,2$ ) satisfies the following equation in the weak sense:

$[ a + b ⁢ ( ∬ ℝ 2 ⁢ N | u ⁢ ( x ) - u ⁢ ( y ) | p | x - y | N + p ⁢ s ⁢ 𝑑 x ⁢ 𝑑 y ) θ - 1 ] ⁢ ( - Δ ) p s ⁢ u = | u | p s * - 2 ⁢ u .$

(ii) For $θ≠N/(N-s⁢p)$ , $a=0$ and $b>0$ , it is easy to see that

$μ 0 = ( b ⁢ S N ⁢ ( θ - 1 ) / p ⁢ s ) - 1 p ⁢ ( θ - 1 ) / ( p s * - p ) - 1$

is a solution of equation (3.1). Thus $vε,μ0=μ01/(ps*-p)⁢Uε$ satisfies the following equation:

$b ⁢ ( ∬ ℝ 2 ⁢ N | u ⁢ ( x ) - u ⁢ ( y ) | p | x - y | N + p ⁢ s ⁢ 𝑑 x ⁢ 𝑑 y ) θ - 1 ⁢ ( - Δ ) p s ⁢ u = | u | p s * - 2 ⁢ u .$

For $θ>N/(N-s⁢p)$ , $a>0$ and $b>0$ define

$g ⁢ ( μ ) = b ⁢ S N ⁢ ( θ - 1 ) / ( p ⁢ s ) ⁢ μ p ⁢ ( θ - 1 ) / ( p s * - p ) - 1 + a ⁢ μ - 1 for ⁢ μ > 0 .$

If a, b and θ satisfy

$a θ ⁢ p - p s * ⁢ ( b ⁢ S N ⁢ ( θ - 1 ) p ⁢ s ) p s * - p = ( θ ⁢ p - p s * ) θ ⁢ p - p s * ⁢ ( p s * - p ) p s * - p ( ( θ - 1 ) ⁢ p ) p ⁢ ( θ - 1 ) ,$

then we deduce

$min μ > 0 ⁡ g ⁢ ( μ ) = g ⁢ ( μ 0 ) = 1 ,$

where

$μ 0 = ( a ⁢ ( p s * - p ) b ⁢ ( θ ⁢ p - p s * ) ⁢ S N ⁢ ( θ - 1 ) / ( p ⁢ s ) ) p s * - p p ⁢ ( θ - 1 ) .$

Hence $vε,μ0=μ01/(ps*-p)⁢Uε$ is a solution of problem (1.1) for each $ε>0$ . If a, b and θ satisfy

$a θ ⁢ p - p s * ⁢ ( b ⁢ S N ⁢ ( θ - 1 ) p ⁢ s ) p s * - p < ( θ ⁢ p - p s * ) θ ⁢ p - p s * ⁢ ( p s * - p ) p s * - p ( ( θ - 1 ) ⁢ p ) p ⁢ ( θ - 1 ) ,$

then $g⁢(μ0)<1$ . Thus there exist $μ1∈(0,μ0)$ and $μ2∈(μ0,∞)$ such that $g⁢(μi)=1$ for $i=1,2$ , that is $μi$ is a solution of (1.3). Hence $vε,μi=μi1/(ps*-p)⁢Uε$ is a weak solution of problem (1.1).

(iii) When $θ<N/(N-s⁢p)$ , i.e. $p⁢(θ-1)/(ps*-p)<1$ , and $a,b>0$ we define

$h ⁢ ( μ ) = b ⁢ S N ⁢ ( θ - 1 ) / ( p ⁢ s ) ⁢ μ p ⁢ ( θ - 1 ) / ( p s * - p ) - μ + a$

for all $μ>0$ . Then we deduce $h⁢(μ)>0$ for μ small enough. Note that $limμ→∞⁡h⁢(μ)=-∞$ . Hence the function h has a zero point $μ0∈(0,∞)$ , that is $μ0$ is a solution of (3.1). Therefore, $vε,μ0=μ01/(ps*-p)⁢Uε$ is a solution of (1.1) for each $ε>0$ . ∎

Proof of Theorem 1.3.

(i) When we have $θ=N/(N-s⁢p)$ , $λ=0$ , $a=0$ , and $b>S-N/(N-s⁢p)=S-θ$ , suppose that $u∈Ds,p⁢(ℝN)∖{0}$ is a solution of (1.1). Then

$b ⁢ ∥ u ∥ ( θ - 1 ) ⁢ p ⁢ ∥ u ∥ p = | u | p s * p s * .$

From this equality and the definition of S we have

$S - θ ⁢ ∥ u ∥ θ ⁢ p < b ⁢ ∥ u ∥ θ ⁢ p = | u | p s * p s * ≤ S - N / ( N - s ⁢ p ) ⁢ ∥ u ∥ p s * = S - θ ⁢ ∥ u ∥ θ ⁢ p ,$

which leads to a contradiction.

(ii) When $θ=N/(N-s⁢p)$ , $λ=0$ , $a>0$ , and $b≥S-θ$ , suppose that $u∈Ds,p⁢(ℝN)∖{0}$ is a solution of problem (1.1). Then

$( a + b ⁢ ∥ u ∥ ( θ - 1 ) ⁢ p ) ⁢ ∥ u ∥ p = | u | p s * p s * .$

From this equality and the definition of S we have

$S - θ ⁢ ∥ u ∥ θ ⁢ p ≤ b ⁢ ∥ u ∥ θ ⁢ p < ( a + b ⁢ ∥ u ∥ ( θ - 1 ) ⁢ p ) ⁢ ∥ u ∥ p = | u | p s * p s * ≤ S - θ ⁢ ∥ u ∥ θ ⁢ p ,$

which is impossible.

(iii) Let $θ=2$ and $a,b>0$ satisfy

$p s * - p p ⁢ ( p ⁢ a 2 ⁢ p - p s * ) ( p s * - 2 ⁢ p ) / ( p s * - p ) ⁢ 1 S p s * / ( p s * - p ) < b .$

Suppose that $u∈Ds,p⁢(ℝN)∖{0}$ is a solution of problem (1.1). Since $N>2⁢s⁢p$ , i.e. $2⁢p>ps*$ , it follows from the Young inequality that

$1 S p s * / p ⁢ ∥ u ∥ p s * = ∥ u ∥ 2 ⁢ p - p s * ⁢ ( p ⁢ a 2 ⁢ p - p s * ) ( 2 ⁢ p - p s * ) / p ⁢ ∥ u ∥ 2 ⁢ p s * - 2 ⁢ p ⁢ ( p ⁢ a 2 ⁢ p - p s * ) ( p s * - 2 ⁢ p ) / p ⁢ 1 S p s * / p$

$≤ a ⁢ ∥ u ∥ p + p s * - p p ⁢ ( p ⁢ a 2 ⁢ p - p s * ) ( p s * - 2 ⁢ p ) / ( p s * - p ) ⁢ 1 S p s * / ( p s * - p ) ⁢ ∥ u ∥ 2 ⁢ p$

$< a ⁢ ∥ u ∥ p + b ⁢ ∥ u ∥ 2 ⁢ p .$

Employing the fractional Sobolev inequality, we get

$1 S p s * / p ⁢ ∥ u ∥ p s * < a ⁢ ∥ u ∥ p + b ⁢ ∥ u ∥ 2 ⁢ p = ∫ ℝ N | u | p s * ⁢ 𝑑 x ≤ 1 S p s * / p ⁢ ∥ u ∥ p s * ,$

which is a contradiction. Thus, we complete the proof. ∎

4 Proof of Theorem 1.5

In this section, we first prove that there exists $λ*>0$ such that for any $λ∈(0,λ*)$ problem (1.1) has a nontrivial nonnegative solution which is a local minimum. In the sequel, without specific statement, we always assume that $1<θ≤N/(N-s⁢p)$ .

Lemma 4.1.

Assume that $a≥0$ , $b>0$ and $θ<N/(N-s⁢p)$ , or $a>0$ , $0<b<S-ps*/p$ and $θ=N/(N-s⁢p)$ . Then there exist $λ0>0$ and $ρ>0$ such that for any $λ∈(0,λ0)$ one has $Iλ⁢(u)≥α$ for some $α>0$ and for all $u∈Ds,p⁢(RN)$ with $∥u∥=ρ$ .

Proof.

For any $u∈Ds,p⁢(ℝN)$ , by the fractional Sobolev inequality, one has

$I λ ⁢ ( u ) = a p ⁢ ∥ u ∥ p + b θ ⁢ p ⁢ ∥ u ∥ θ ⁢ p - 1 p s * ⁢ ∫ ℝ N | u | p s * ⁢ 𝑑 x - λ ⁢ ∫ ℝ N f ⁢ u ⁢ 𝑑 x$

$≥ a p ⁢ ∥ u ∥ p + b θ ⁢ p ⁢ ∥ u ∥ θ ⁢ p - 1 p s * ⁢ S - p s * / p ⁢ ∥ u ∥ p s * - λ ⁢ S - 1 / p ⁢ | f | ( p s * ) ′ ⁢ ∥ u ∥ .$

When $a≥0$ , $b>0$ and $θ<N/(N-s⁢p)$ , we have

$I λ ⁢ ( u ) ≥ ( b θ ⁢ p ⁢ ∥ u ∥ θ ⁢ p - 1 - 1 p s * ⁢ S - p s * / p ⁢ ∥ u ∥ p s * - 1 - λ ⁢ S - 1 / p ⁢ | f | ( p s * ) ′ ) ⁢ ∥ u ∥ .$

Define

$g ⁢ ( t ) := b θ ⁢ p ⁢ t θ ⁢ p - 1 - 1 p s * ⁢ S - p s * / p ⁢ t p s * - 1$

for all $t≥0$ . Since $ps*>θ⁢p$ by $θ<N/(N-s⁢p)$ , we obtain that $maxt≥0⁡g⁢(t)=g⁢(ρ)>0$ , where

$ρ = ( b ⁢ S p s * / p ⁢ ( θ ⁢ p - 1 ) ⁢ p s * θ ⁢ p ⁢ ( p s * - 1 ) ) 1 / ( p s * - θ ⁢ p ) .$

Set

$λ 0 = g ⁢ ( ρ ) S - 1 / p ⁢ | f | ( p s * ) ′ .$

Then for any $λ∈(0,λ0)$ we get

$I λ ( u ) ≥ ( g ( ρ ) - λ S - 1 / p | f | ( p s * ) ′ ) ρ = : α > 0 for all ∥ u ∥ = ρ .$

When $θ=N/(N-s⁢p)$ , $a>0$ and $0<b<S-ps*/p$ , we have

$I λ ⁢ ( u ) ≥ [ a p ⁢ ∥ u ∥ p - 1 - ( 1 p s * ⁢ S - p s * / p - b p s * ) ⁢ ∥ u ∥ θ ⁢ p - 1 - λ ⁢ S - 1 / p ⁢ | f | ( p s * ) ′ ] ⁢ ∥ u ∥$

with $ps*=θ⁢p$ . Set

$h ⁢ ( t ) := a p ⁢ t p - 1 - ( 1 p s * ⁢ S - p s * / p - b θ ⁢ p ) ⁢ t θ ⁢ p - 1$

for all $t≥0$ . Then $maxt≥0⁡h⁢(t)=h⁢(ρ)>0$ , where

$ρ = ( ( p - 1 ) ⁢ p s * ⁢ a p ⁢ ( p s * - 1 ) ⁢ ( S - p s * / p - b ) ) 1 / ( p s * - p ) .$

Let

$λ 0 = h ⁢ ( ρ ) S - 1 / p ⁢ | f | ( p s * ) ′ .$

Then for all $λ∈(0,λ0)$ we have

$I λ ( u ) ≥ ( h ( ρ ) - λ S - 1 / p | f | ( p s * ) ′ ) ρ = : α > 0 for all ∥ u ∥ = ρ .$

Thus, we complete the proof. ∎

Lemma 4.2.

Assume that $θ<N/(N-s⁢p)$ , $a≥0$ and $b>0$ , or $θ=N/(N-s⁢p)$ , $a>0$ and $b<S-θ$ . Suppose that $λ>0$ . Then there exists $e∈Ds,p⁢(RN)$ such that $∥e∥≥ρ$ and $Iλ⁢(e)<0$ , where ρ is the number given in Lemma 4.1.

Proof.

For $U1$ in (1.5), we have $∥U1∥p=∥U1∥ps*ps*=SNs⁢p$ . Then for any $t>0$ we have

$I λ ⁢ ( t ⁢ U 1 ) = a ⁢ t p ⁢ S N s ⁢ p p + b ⁢ t θ ⁢ p ⁢ S N ⁢ θ s ⁢ p θ ⁢ p - t p s * ⁢ S N s ⁢ p p s * - λ ⁢ t ⁢ ∫ ℝ N f ⁢ U 1 ⁢ 𝑑 x$

$≤ a ⁢ t p ⁢ S N s ⁢ p p + b ⁢ t θ ⁢ p ⁢ S N ⁢ θ s ⁢ p θ ⁢ p - t p s * ⁢ S N s ⁢ p p s * .$

When $θ<N/(N-s⁢p)$ , i.e. $ps*>θ⁢p$ , we obtain that there exists $t>0$ large enough such that $∥t⁢U1∥≥ρ$ and $Iλ⁢(t⁢U1)<0$ . When $θ=N/(N-s⁢p)$ , we obtain

$I λ ⁢ ( t ⁢ U 1 ) ≤ a ⁢ t p ⁢ S N s ⁢ p p - ( S N s ⁢ p p s * - b ⁢ S N ⁢ θ s ⁢ p θ ⁢ p ) ⁢ t p s *$

$= a ⁢ t p ⁢ S N s ⁢ p p - ( S θ θ - 1 p s * - b ⁢ S θ 2 θ - 1 p s * ) ⁢ t p s * .$

It follows from $b<S-θ$ that there exists $t>0$ such that $∥t⁢U1∥≥ρ$ and $Iλ⁢(t⁢U1)<0$ . Then the result follows by taking $e=t⁢U1$ . ∎

Definition 4.3.

A sequence ${un}n⊂Ds,p⁢(ℝN)$ is called a $(PS)c$ sequence if $Iλ⁢(un)→c$ and $Iλ′⁢(un)→0$ .

Lemma 4.4.

If ${un}n$ is a $(PS)c$ sequence, then ${un}n$ is bounded and ${un+}n$ is also a $(PS)c$ sequence.

Proof.

Since ${un}n$ is a $(PS)c$ sequence, there exist $n0>0$ and $C>0$ such that

(4.1)

$I λ ⁢ ( u n ) - 1 p s * ⁢ 〈 I λ ′ ⁢ ( u n ) , u n 〉 ≤ C + C ⁢ ∥ u n ∥ for all ⁢ n ≥ n 0 .$

When $θ<N/(N-s⁢p)$ , $a≥0$ and $b>0$ , we have

$I λ ⁢ ( u n ) - 1 p s * ⁢ 〈 I λ ′ ⁢ ( u n ) , u n 〉 = a ⁢ ( 1 p - 1 p s * ) ⁢ ∥ u n ∥ p + b ⁢ ( 1 θ ⁢ p - 1 p s * ) ⁢ ∥ u n ∥ θ ⁢ p + λ ⁢ ( 1 p s * - 1 ) ⁢ ∫ ℝ N f ⁢ ( x ) ⁢ u n ⁢ 𝑑 x$

$≥ b ⁢ ( 1 θ ⁢ p - 1 p s * ) ⁢ ∥ u n ∥ θ ⁢ p - λ ⁢ ( 1 - 1 p s * ) ⁢ S - 1 / p ⁢ | f | ( p s * ) ′ ⁢ ∥ u n ∥ .$

When $θ=N/(N-s⁢p)$ , $a>0$ and $b>0$ , we have

$≥ a ⁢ ( 1 p - 1 p s * ) ⁢ ∥ u n ∥ p - λ ⁢ ( 1 - 1 p s * ) ⁢ S - 1 / p ⁢ | f | ( p s * ) ′ ⁢ ∥ u n ∥ .$

Using $θ⁢p>p>1$ and (4.1), we obtain that ${un}n$ is bounded in $Ds,p⁢(ℝN)$ .

Now we shall show that ${un+}n$ is also a $(PS)c$ sequence. Since ${un}n$ is bounded, the sequence $un-=un-un+$ is also bounded. Then $〈Iλ′⁢(un),-un-〉→0$ and it follows that

$( a + b ⁢ ∥ u n ∥ ( θ - 1 ) ⁢ p ) ⁢ ∬ ℝ 2 ⁢ N | u n ⁢ ( x ) - u n ⁢ ( y ) | p - 2 ⁢ ( u n ⁢ ( x ) - u n ⁢ ( y ) ) ⁢ ( - u n - ⁢ ( x ) + u n - ⁢ ( y ) ) | x - y | N + p ⁢ s ⁢ 𝑑 x ⁢ 𝑑 y = λ ⁢ ∫ ℝ N f ⁢ ( - u n - ) ⁢ 𝑑 x + o ⁢ ( 1 ) .$

From the fact that $f≥0$ we have

$( a + b ⁢ ∥ u n ∥ ( θ - 1 ) ⁢ p ) ⁢ ∬ ℝ 2 ⁢ N | u n ⁢ ( x ) - u n ⁢ ( y ) | p - 2 ⁢ ( u n ⁢ ( x ) - u n ⁢ ( y ) ) ⁢ ( - u n - ⁢ ( x ) + u n - ⁢ ( y ) ) | x - y | N + p ⁢ s ⁢ 𝑑 x ⁢ 𝑑 y = o ⁢ ( 1 ) .$

From this, together with the facts

$| ξ - - η - | p ≤ | ξ - η | p - 2 ⁢ ( ξ - η ) ⁢ ( - ξ - + η - )$

and

$| ξ - - η - | ≤ | ξ - η |$

for all $ξ,η∈ℝ$ , we obtain

$( a + b ⁢ ∥ u n - ∥ ( θ - 1 ) ⁢ p ) ⁢ ∬ ℝ 2 ⁢ N | u n - ⁢ ( x ) - u n - ⁢ ( y ) | p | x - y | N + p ⁢ s ⁢ 𝑑 x ⁢ 𝑑 y = o ⁢ ( 1 ) .$

Moreover, by $a≥0$ and $b>0$ , we get

$b ⁢ ∥ u n - ∥ θ ⁢ p = o ⁢ ( 1 ) .$

Hence, $∥un-∥→0$ as $n→∞$ . This together with the fractional Sobolev inequality implies that $|un-|ps*→0$ as $n→∞$ .

Note that

$| ∬ ℝ 2 ⁢ N | u n ⁢ ( x ) - u n ⁢ ( y ) | p - | u n + ⁢ ( x ) - u n + ⁢ ( y ) | p | x - y | N + p ⁢ s ⁢ 𝑑 x ⁢ 𝑑 y |$

$≤ p ⁢ ∬ ℝ 2 ⁢ N | | u n + ⁢ ( x ) - u n + ⁢ ( y ) | + κ 1 ⁢ ( | u n ⁢ ( x ) - u n ⁢ ( y ) | - | u n + ⁢ ( x ) - u n + ⁢ ( y ) | ) | p - 1 | x - y | N + p ⁢ s$

$≤ p ⁢ 2 p - 1 ⁢ ∬ ℝ 2 ⁢ N | u n ⁢ ( x ) - u n ⁢ ( y ) | p - 1 | x - y | N + p ⁢ s ⁢ | u n - ⁢ ( x ) - u n - ⁢ ( y ) | ⁢ 𝑑 x ⁢ 𝑑 y$

$≤ p ⁢ 2 p - 1 ⁢ ∥ u n ∥ p - 1 ⁢ ∥ u n - ∥ → 0 as ⁢ n → ∞ ,$

where $κ1∈(0,1)$ . Thus, we arrive at

(4.2)

$∥ u n ∥ p = ∥ u n + ∥ p + o ⁢ ( 1 ) .$

Similarly,

(4.3)

$| u n | p s * p s * = | u n + | p s * p s * + o ⁢ ( 1 ) .$

Note also that

$| ∥ u n ∥ θ ⁢ p - ∥ u n + ∥ θ ⁢ p | = θ ⁢ | ∥ u n + ∥ θ ⁢ p + κ 2 ⁢ ( ∥ u n ∥ p - ∥ u n + ∥ p ) | θ - 1 ⁢ | ∥ u n ∥ p - ∥ u n + ∥ p |$

(4.4)

$≤ C ⁢ | ∥ u n ∥ p - ∥ u n + ∥ p | → 0$

as $n→∞$ , where $κ2∈(0,1)$ and $C>0$ is a constant independent of n. Therefore, by equations (4.2) and (4.3) and inequality (4.4), we obtain $Iλ⁢(un)=Iλ⁢(un+)+o⁢(1)$ and $Iλ′⁢(un)=Iλ′⁢(un+)+o⁢(1)$ . Consequently, we get that ${un+}n$ is a $(PS)c$ sequence. ∎

It follows from Lemma 4.4 that any $(PS)c$ sequence can be regarded as a sequence of nonnegative functions. Thus, in the following we always assume that a $(PS)c$ sequence is nonnegative.

Set

$Λ = { b p s * p s * - θ ⁢ p ⁢ ( 1 θ ⁢ p - 1 p s * ) ⁢ S N ⁢ θ s ⁢ p ⁢ p s * - p p s * - θ ⁢ p if ⁢ θ < N N - s ⁢ p , a = 0 , b > 0 , ( b ⁢ S 2 ⁢ θ - 1 2 + b 2 ⁢ S 2 ⁢ θ - 1 + 4 ⁢ a 2 ) 1 θ - 1 ⁢ a ⁢ ( 1 p - 1 p s * ) ⁢ S N s ⁢ p + b ⁢ ( 1 θ ⁢ p - 1 p s * ) ⁢ ( b ⁢ S 2 ⁢ θ - 1 2 + b 2 ⁢ S 2 ⁢ θ - 1 + 4 ⁢ a 2 ) θ θ - 1 ⁢ S N ⁢ θ s ⁢ p if ⁢ θ = N - s ⁢ p / 2 N - s ⁢ p , a , b > 0 , ( 1 p - 1 θ ⁢ p ) ⁢ ( a 1 - b ⁢ S θ ) θ θ - 1 ⁢ S θ θ - 1 if ⁢ θ = N N - s ⁢ p , a > 0 , 0 < b < S - θ ,$

and

$Θ ⁢ ( λ ) = { ( a ⁢ S p - a ⁢ S θ ⁢ p ) 1 - p ′ ⁢ ( 1 - 1 θ ⁢ p ) p ′ ⁢ λ p ′ ⁢ | f | ( p s * ) ′ p ′ if ⁢ a > 0 , ( 1 θ ⁢ p - 1 p s * ) 1 - ( p s * ) ′ ⁢ ( 1 - 1 θ ⁢ p ) ( p s * ) ′ ⁢ λ ( p s * ) ′ ⁢ | f | ( p s * ) ′ ( p s * ) ′ if ⁢ a = 0$

for $λ>0$ , where $p′=p/(p-1)$ . Obviously, $Θ⁢(λ)$ is increasing in $(0,∞)$ . Then we have the following result.

Lemma 4.5.

Suppose that $θ<N/(N-s⁢p)$ , $a≥0$ and $b>0$ , or $θ=N/(N-s⁢p)$ , $a>0$ and $0<b<S-θ$ . If ${un}n⊂Ds,p⁢(RN)$ is a bounded and nonnegative sequence which satisfies $Iλ⁢(un)→c<Λ-Θ⁢(λ)$ and $Iλ′⁢(un)→0$ in $(Ds,p⁢(RN))′$ , then there exists a nonnegative function $u∈Ds,p⁢(RN)$ such that, up to a subsequence, $un→u$ in $Ds,p⁢(RN)$ .

Proof.

Since ${un}n⊂Ds,p⁢(ℝN)$ is bounded and nonnegative, up to a subsequence, there exists a nonnegative function $u∈Ds,p⁢(ℝN)$ such that $un⇀u$ in $Ds,p⁢(ℝN)$ , $un→u$ in $Llocσ$ for $σ∈[1,ps*)$ and $un→u$ a.e. in $ℝN$ . By Theorem 2.2, up to a subsequence, there exist a (at most) countable set J, a non-atomic measure $μ~$ and points ${xj}j∈J⊂ℝN$ and ${μj}j∈J,{νj}j∈J⊂ℝ+$ such that as $n→∞$ we have

(4.5)

$∫ ℝ N | u n ⁢ ( x ) - u n ⁢ ( y ) | p | x - y | N + p ⁢ s ⁢ 𝑑 y ⇀ μ = ∫ ℝ N | u ⁢ ( x ) - u ⁢ ( y ) | p | x - y | N + p ⁢ s ⁢ 𝑑 y + ∑ j ∈ J μ j ⁢ δ x j + μ ~$

and

(4.6)

$| u n | p s * ⇀ ν = | u | p s * + ∑ j ∈ J ν j ⁢ δ x j$

in the measure sense, where $δxj$ is the Dirac measure concentrated at $xj$ . Moreover,

(4.7)

$ν j ≤ S - p s * / p ⁢ μ j p s * / p for all ⁢ j ∈ J ,$

and $S>0$ is the best constant of $Ds,p⁢(ℝN)↪Lps*⁢(ℝN)$ .

Next we claim that $J=∅$ . Suppose by contradiction that $J≠∅$ . Fix $j∈J$ . For $ε>0$ choose $φε,j∈C0∞⁢(ℝN)$ such that

$φ ε , j = 1 ⁢ for ⁢ | x - x j | ≤ ε , φ ε , j = 0 ⁢ for ⁢ | x - x j | ≥ 2 ⁢ ε ,$

and $|∇⁡φε,j|≤2/ε$ . Obviously, $φε,j⁢un∈Ds.p⁢(ℝN)$ . It follows from $〈Iλ′⁢(un),φε,j⁢un〉→0$ that

$( a + b ⁢ ∥ u n ∥ ( θ - 1 ) ⁢ p ) ⁢ ∬ ℝ 2 ⁢ N | u n ⁢ ( x ) - u n ⁢ ( y ) | p - 2 ⁢ ( u n ⁢ ( x ) - u n ⁢ ( y ) ) ⁢ ( φ ε , j ⁢ ( x ) ⁢ u n ⁢ ( x ) - φ ε , j ⁢ ( y ) ⁢ u n ⁢ ( y ) ) | x - y | N + p ⁢ s ⁢ 𝑑 x ⁢ 𝑑 y$

(4.8)

$= ∫ ℝ N u n p s * ⁢ φ ε , j ⁢ 𝑑 x + λ ⁢ ∫ ℝ N f ⁢ φ ε , j ⁢ u n ⁢ 𝑑 x + o ⁢ ( 1 ) .$

Using the Hölder inequality and Lemma 2.3, we deduce

$≤ C ⁢ lim ε → 0 ⁡ lim n → ∞ ⁡ ( ∬ ℝ 2 ⁢ N | u n ⁢ ( x ) - u n ⁢ ( y ) | p | x - y | N + p ⁢ s ⁢ 𝑑 x ⁢ 𝑑 y ) ( p - 1 ) / p ⁢ ( ∬ ℝ 2 ⁢ N | ( φ ε , j ⁢ ( x ) - φ ε , j ⁢ ( y ) ) ⁢ u n ⁢ ( x ) | p | x - y | N + p ⁢ s ⁢ 𝑑 x ⁢ 𝑑 y ) 1 / p$

(4.9)

$≤ C ⁢ lim ε → 0 ⁡ lim n → ∞ ⁡ ( ∬ ℝ 2 ⁢ N | ( φ ε , j ⁢ ( x ) - φ ε , j ⁢ ( y ) ) ⁢ u n ⁢ ( x ) | p | x - y | N + p ⁢ s ⁢ 𝑑 x ⁢ 𝑑 y ) 1 / p = 0 .$

By (4.5) and (4.6), we have

$lim ε → 0 ⁡ lim n → ∞ ⁡ ( a + b ⁢ ∥ u n ∥ ( θ - 1 ) ⁢ p ) ⁢ ∬ ℝ 2 ⁢ N | u n ⁢ ( x ) - u n ⁢ ( y ) | p | x - y | N + p ⁢ s ⁢ φ ε , j ⁢ ( y ) ⁢ 𝑑 y ⁢ 𝑑 x$

$≥ lim ε → 0 lim n → ∞ [ a ∬ ℝ 2 ⁢ N | u n ⁢ ( x ) - u n ⁢ ( y ) | p | x - y | N + p ⁢ s φ ε , j ( y ) d x d y + b ( ∬ ℝ 2 ⁢ N | u n ⁢ ( x ) - u n ⁢ ( y ) | p | x - y | N + p ⁢ s φ ε , j ( y ) d x d y ) θ ]$

(4.10)

$= a ⁢ μ j + b ⁢ μ j θ$

and

(4.11)

$lim ε → 0 ⁡ lim n → ∞ ⁡ ∫ ℝ N u n p s * ⁢ φ ε , j ⁢ 𝑑 x = lim ε → 0 ⁡ ∫ ℝ N u p s * ⁢ φ ε , j ⁢ 𝑑 x + ν j = ν j ,$

and furthermore

(4.12)

$lim ε → 0 ⁡ lim n → ∞ ⁡ ∫ ℝ N f ⁢ φ ε , j ⁢ u n ⁢ 𝑑 x = lim ε → 0 ⁡ ∫ ℝ N f ⁢ φ ε , j ⁢ u ⁢ 𝑑 x = 0 .$

It follows from (4.8)–(4.12) that

$ν j ≥ a ⁢ μ j + b ⁢ μ j θ .$

Combining this inequality with (4.7), we obtain

(4.13)

$ν j ≥ { ( a ⁢ S 1 - b ⁢ S θ ) θ θ - 1 if ⁢ θ = N N - s ⁢ p , a > 0 , 0 < b < S - θ , ( b ⁢ S θ ) p s * p s * - θ ⁢ p if ⁢ θ < N N - s ⁢ p , a = 0 , b > 0 , ( b ⁢ S θ + b 2 ⁢ S 2 ⁢ θ + 4 ⁢ a ⁢ S 2 ) 2 ⁢ θ - 1 θ - 1 if ⁢ θ = N - s ⁢ p / 2 N - s ⁢ p , a , b > 0 .$

For $R>0$ assume that $ψR∈C0∞⁢(ℝN)$ satisfies $ψR∈[0,1]$ , $ψR⁢(x)=1$ for $|x-xj|≤R$ , $ψR⁢(x)=0$ for $|x-xj|>2⁢R$ , and $|∇⁡ψR|≤2/R$ . By (4.5) and (4.6), we obtain that

$c = lim n → ∞ ⁡ ( I λ ⁢ ( u n ) - 1 θ ⁢ p ⁢ 〈 I λ ′ ⁢ ( u n ) , u n 〉 )$

$≥ lim R → ∞ lim n → ∞ ( ( a p - a θ ⁢ p ) ∬ ℝ 2 ⁢ N | u n ⁢ ( x ) - u n ⁢ ( y ) | p | x - y | N + p ⁢ s ψ R ( x ) d x d y + ( a p - a θ ⁢ p ) μ j$

$+ ( 1 θ ⁢ p - 1 p s * ) ∫ ℝ N u n p s * ψ R d x - ( 1 - 1 θ ⁢ p ) λ ∫ ℝ N f u n d x )$

$≥ lim R → ∞ ( ( a p - a θ ⁢ p ) ∬ ℝ 2 ⁢ N | u ⁢ ( x ) - u ⁢ ( y ) | p | x - y | N + p ⁢ s ψ R ( x ) d x d y + ( a p - a θ ⁢ p ) μ j$

$+ ( 1 θ ⁢ p - 1 p s * ) ∫ ℝ N u p s * ψ R d x + ( 1 θ ⁢ p - 1 p s * ) ν j - ( 1 - 1 θ ⁢ p ) λ ∫ ℝ N f u d x )$

$≥ ( a p - a θ ⁢ p ) ⁢ ∬ ℝ 2 ⁢ N | u ⁢ ( x ) - u ⁢ ( y ) | p | x - y | N + p ⁢ s ⁢ 𝑑 x ⁢ 𝑑 y + ( a p - a θ ⁢ p ) ⁢ μ j$

$+ ( 1 θ ⁢ p - 1 p s * ) ⁢ ∫ ℝ N u p s * ⁢ 𝑑 x + ( 1 θ ⁢ p - 1 p s * ) ⁢ ν j - ( 1 - 1 θ ⁢ p ) ⁢ λ ⁢ ∫ ℝ N f ⁢ u ⁢ 𝑑 x$

$≥ ( a p - a θ ⁢ p ) ⁢ S ⁢ | u | p s * p + ( a p - a θ ⁢ p ) ⁢ μ j + ( 1 θ ⁢ p - 1 p s * ) ⁢ ∫ ℝ N u p s * ⁢ 𝑑 x + ( 1 θ ⁢ p - 1 p s * ) ⁢ ν j - ( 1 - 1 θ ⁢ p ) ⁢ λ ⁢ ∫ ℝ N f ⁢ u ⁢ 𝑑 x ,$

which implies that

$c ≥ { ( a p - a θ ⁢ p ) ⁢ S ⁢ | u | p s * p + ( a p - a θ ⁢ p ) ⁢ μ j + ( 1 θ ⁢ p - 1 p s * ) ⁢ ν j - ( 1 - 1 θ ⁢ p ) ⁢ λ ⁢ ∫ ℝ N f ⁢ u ⁢ 𝑑 x if ⁢ θ = N N - s ⁢ p , a > 0 , b > 0 , ( 1 θ ⁢ p - 1 p s * ) ⁢ ∫ ℝ N u p s * ⁢ 𝑑 x + ( 1 θ ⁢ p - 1 p s * ) ⁢ ν j - ( 1 - 1 θ ⁢ p ) ⁢ λ ⁢ ∫ ℝ N f ⁢ u ⁢ 𝑑 x if ⁢ θ < N N - s ⁢ p , a = 0 , b > 0 , ( a p - a θ ⁢ p ) ⁢ S ⁢ | u | p s * p + ( a p - a θ ⁢ p ) ⁢ μ j + ( 1 θ ⁢ p - 1 p s * ) ⁢ ν j - ( 1 - 1 θ ⁢ p ) ⁢ λ ⁢ ∫ ℝ N f ⁢ u ⁢ 𝑑 x if ⁢ θ = N - s ⁢ p / 2 N - s ⁢ p , a > 0 , b > 0 .$

By the Hölder and Young inequalities, we have

$( 1 - 1 θ ⁢ p ) ⁢ λ ⁢ ∫ ℝ N f ⁢ u ⁢ 𝑑 x ≤ ( 1 - 1 θ ⁢ p ) ⁢ λ ⁢ | f | p s * p s * - 1 ⁢ | u | p s *$

(4.14)

$≤ ( a p - a θ ⁢ p ) ⁢ S ⁢ | u | p s * p + ( a ⁢ S p - a ⁢ S θ ⁢ p ) 1 - p ′ ⁢ ( 1 - 1 θ ⁢ p ) p ′ ⁢ λ p ′ ⁢ | f | ( p s * ) ′ p ′$

for $a>0$ and

$( 1 - 1 θ ⁢ p ) ⁢ λ ⁢ ∫ ℝ N f ⁢ u ⁢ 𝑑 x ≤ ( 1 - 1 θ ⁢ p ) ⁢ λ ⁢ | f | ( p s * ) ′ ⁢ | u | p s *$

(4.15)

$≤ ( 1 θ ⁢ p - 1 p s * ) ⁢ | u | p s * p s * + ( 1 θ ⁢ p - 1 p s * ) 1 - ( p s * ) ′ ⁢ ( 1 - 1 θ ⁢ p ) ( p s * ) ′ ⁢ λ ( p s * ) ′ ⁢ | f | ( p s * ) ′ ( p s * ) ′$

for $a=0$ , where $p′=p/(p-1)$ . Hence we obtain by (4.14) and (4.15) that

$c ≥ { ( a p - a θ ⁢ p ) ⁢ μ j - ( a ⁢ S p - a ⁢ S θ ⁢ p ) 1 - p ′ ⁢ ( 1 - 1 θ ⁢ p ) p ′ ⁢ λ p ′ ⁢ | f | ( p s * ) ′ p ′ if ⁢ θ = N N - s ⁢ p , a > 0 , b > 0 , ( 1 θ ⁢ p - 1 p s * ) ⁢ ν j - ( 1 θ ⁢ p - 1 p s * ) 1 - ( p s * ) ′ ⁢ ( 1 - 1 θ ⁢ p ) ( p s * ) ′ ⁢ λ ( p s * ) ′ ⁢ | f | ( p s * ) ′ ( p s * ) ′ if ⁢ θ < N N - s ⁢ p , a = 0 , b > 0 , ( a p - a p s * ) ⁢ μ j + b ⁢ ( 1 θ ⁢ p - 1 p s * ) ⁢ μ j θ - ( a ⁢ S p - a ⁢ S θ ⁢ p ) 1 - p ′ ⁢ ( 1 - 1 θ ⁢ p ) p ′ ⁢ λ p ′ ⁢ | f | ( p s * ) ′ p ′ if ⁢ θ = N - s ⁢ p / 2 N - s ⁢ p , a > 0 , b > 0 ,$

where the third inequality follows by using $νj≥a⁢μj+b⁢μjθ$ . Combining (4.7) and (4.13), we get $c≥Λ-Θ⁢(λ)$ , which is a contradiction. Hence the claim holds.

Let $R>0$ . We define

$μ ∞ = lim R → ∞ ⁡ lim sup n → ∞ ⁡ ∫ { x ∈ ℝ N : | x | > R } ∫ ℝ N | u n ⁢ ( x ) - u n ⁢ ( y ) | p | x - y | N + p ⁢ s ⁢ 𝑑 y ⁢ 𝑑 x$

and

$ν ∞ = lim R → ∞ ⁡ lim sup n → ∞ ⁡ ∫ { x ∈ ℝ N : | x | > R } | u n | p s * ⁢ 𝑑 x .$

It follows from Theorem 2.4 that $μ∞$ and $ν∞$ are well defined and they satisfy

(4.16)

$lim sup n → ∞ ⁡ ∫ ℝ N ∫ ℝ N | u n ⁢ ( x ) - u n ⁢ ( y ) | p | x - y | N + p ⁢ s ⁢ 𝑑 y ⁢ 𝑑 x = ∫ ℝ N 𝑑 μ + μ ∞$

and

(4.17)

$lim sup n → ∞ ⁡ ∫ ℝ N u n p s * ⁢ 𝑑 x = ∫ ℝ N 𝑑 ν + ν ∞ .$

Assume that $χR∈C∞⁢(ℝℕ)$ satisfies $χR∈[0,1]$ and $χR⁢(x)=0$ for $|x|<R$ , $χR⁢(x)=1$ for $|x|>2⁢R$ , and $|∇⁡χR|≤2/R$ . With a similar discussion as in the proof of Theorem 2.4, we have

(4.18)

$μ ∞ = lim R → ∞ ⁡ lim sup n → ∞ ⁡ ∬ ℝ N | u n ⁢ ( x ) - u n ⁢ ( y ) | p ⁢ χ R ⁢ ( x ) | x - y | N + p ⁢ s ⁢ 𝑑 y ⁢ 𝑑 x$

and

(4.19)

$ν ∞ = lim R → ∞ ⁡ lim sup n → ∞ ⁡ ∫ ℝ N | u n ⁢ ( x ) ⁢ χ R ⁢ ( x ) | p s * ⁢ 𝑑 x .$

Moreover, we have

(4.20)

$S ⁢ ν ∞ p / p s * ≤ μ ∞ .$

Since $∥un∥p$ and $∥un∥ps*ps*$ are bounded, up to a subsequence, we can assume that $∥un∥p$ and $∥un∥ps*ps*$ are both convergent. Then by (4.16) and (4.17) we obtain

(4.21)

$lim n → ∞ ⁡ ∥ u n ∥ p = ∫ ℝ N 𝑑 μ + μ ∞$

and

(4.22)

$lim n → ∞ ⁡ ∥ u n ∥ p s * p s * = ∫ ℝ N 𝑑 ν + ν ∞ .$

It follows from $〈Iλ′⁢(un),χR⁢un〉→0$ as $n→∞$ that

$( a + b ∥ u n ∥ ( θ - 1 ) ⁢ p ) [ ∬ ℝ 2 ⁢ N | u n ⁢ ( x ) - u n ⁢ ( y ) | p ⁢ χ R ⁢ ( x ) | x - y | N + p ⁢ s d x d y$

$+ ∬ ℝ 2 ⁢ N | u n ⁢ ( x ) - u n ⁢ ( y ) | p - 2 ⁢ ( u n ⁢ ( x ) - u n ⁢ ( y ) ) ⁢ u n ⁢ ( y ) ⁢ ( χ R ⁢ ( x ) - χ R ⁢ ( y ) ) | x - y | N + p ⁢ s d x d y ]$

(4.23)

$= ∫ ℝ N u n p s * ⁢ χ R ⁢ 𝑑 x + λ ⁢ ∫ ℝ N f ⁢ u n ⁢ χ R ⁢ 𝑑 x + o ⁢ ( 1 ) .$

By (2.15) and the Hölder inequality, we get

(4.24)

$lim R → ∞ ⁡ lim sup n → ∞ ⁡ ∬ ℝ 2 ⁢ N | u n ⁢ ( x ) - u n ⁢ ( y ) | p - 2 ⁢ ( u n ⁢ ( x ) - u n ⁢ ( y ) ) ⁢ u n ⁢ ( y ) ⁢ ( χ R ⁢ ( x ) - χ R ⁢ ( y ) ) | x - y | N + p ⁢ s ⁢ 𝑑 x ⁢ 𝑑 y = 0 .$

Hence we deduce from (4.18), (4.21), (4.23) and (4.24) that

$lim R → ∞ ⁡ lim sup n → ∞ ⁡ ( a + b ⁢ ∥ u n ∥ ( θ - 1 ) ⁢ p ) ⁢ ∬ ℝ 2 ⁢ N | u n ⁢ ( x ) - u n ⁢ ( y ) | p ⁢ χ R ⁢ ( x ) | x - y | N + p ⁢ s ⁢ 𝑑 x ⁢ 𝑑 y$

$= [ a + b ⁢ ( ∫ ℝ N 𝑑 μ + μ ∞ ) θ - 1 ] ⁢ lim R → ∞ ⁡ lim sup n → ∞ ⁡ ( ∫ { x ∈ ℝ N : | x | > R } ∫ ℝ N | u n ⁢ ( x ) - u n ⁢ ( y ) | p | x - y | N + p ⁢ s ⁢ 𝑑 y ⁢ 𝑑 x )$

$≥ ( a + b ⁢ μ ∞ θ - 1 ) ⁢ μ ∞$

(4.25)

$= a ⁢ μ ∞ + b ⁢ μ ∞ θ ,$

thanks to $θ>1$ . It is easy to see that

(4.26)

$lim R → ∞ ⁡ lim sup n → ∞ ⁡ ∫ ℝ N f ⁢ χ R ⁢ u n ⁢ 𝑑 x = lim R → ∞ ⁡ ∫ ℝ N f ⁢ χ R ⁢ u ⁢ 𝑑 x = 0 .$

Therefore, we conclude from (4.23)–(4.26) and (4.19) that

$a ⁢ μ ∞ + b ⁢ μ ∞ θ ≤ ν ∞ .$

This together with (4.20) yields

$a ⁢ S ⁢ ν ∞ p / p s * + b ⁢ S θ ⁢ ν ∞ θ ⁢ p / p s * ≤ ν ∞ ,$

which implies that $ν∞=0$ or

(4.27)

$ν ∞ ≥ { ( a ⁢ S 1 - b ⁢ S θ ) θ θ - 1 if ⁢ θ = N N - s ⁢ p , a > 0 , 0 < b < S - θ , ( b ⁢ S θ ) p s * p s * - θ ⁢ p if ⁢ θ < N N - s ⁢ p , a = 0 , b > 0 , ( b ⁢ S θ + b 2 ⁢ S 2 ⁢ θ + 4 ⁢ a ⁢ S 2 ) 2 ⁢ θ - 1 θ - 1 if ⁢ θ = N - s ⁢ p / 2 N - s ⁢ p , a , b > 0 .$

Assume that (4.27) holds. Then

$c = lim n → ∞ ⁡ ( I λ ⁢ ( u n ) - 1 θ ⁢ p ⁢ 〈 I λ ′ ⁢ ( u n ) , u n 〉 )$

$= lim n → ∞ ⁡ ( ( a p - a θ ⁢ p ) ⁢ ∥ u n ∥ p + ( 1 θ ⁢ p - 1 p s * ) ⁢ ∫ ℝ N u n p s * ⁢ 𝑑 x - ( 1 - 1 θ ⁢ p ) ⁢ λ ⁢ ∫ ℝ N f ⁢ u n )$

$= ( a p - a θ ⁢ p ) ⁢ ∫ ℝ N 𝑑 μ + ( 1 p - 1 θ ⁢ p ) ⁢ μ ∞ + ( 1 θ ⁢ p - 1 p s * ) ⁢ ∫ ℝ N 𝑑 ν + ( 1 θ ⁢ p - 1 p s * ) ⁢ ν ∞ - ( 1 - 1 θ ⁢ p ) ⁢ λ ⁢ ∫ ℝ N f ⁢ u ⁢ 𝑑 x$

$≥ ( a p - a θ ⁢ p ) ⁢ ∥ u ∥ p + ( 1 p - 1 θ ⁢ p ) ⁢ μ ∞ + ( 1 θ ⁢ p - 1 p s * ) ⁢ ν ∞ - ( 1 - 1 θ ⁢ p ) ⁢ λ ⁢ ∫ ℝ N f ⁢ u ⁢ 𝑑 x .$

Thus, we deduce from (4.14), (4.15) and (4.27) that $c≥Λ-Θ⁢(λ)$ , which is a contradiction. Hence, $ν∞=0$ . In view of $J=∅$ and (4.22), we have

(4.28)

$lim n → ∞ ⁡ ∫ ℝ N u n p s * ⁢ 𝑑 x = ∫ ℝ N u p s * ⁢ 𝑑 x .$

Now we show that $un→u$ in $Ds,p⁢(ℝN)$ . To this end, we first assume that $d:=infn≥1⁡∥un∥>0$ .

For any $w,v∈Ds,p⁢(ℝN)$ we define

$[ w , v ] = ∬ ℝ 2 ⁢ N | w ⁢ ( x ) - w ⁢ ( y ) | p - 2 ⁢ ( w ⁢ ( x ) - w ⁢ ( y ) ) ⁢ ( v ⁢ ( x ) - v ⁢ ( y ) ) | x - y | N + p ⁢ s ⁢ 𝑑 x ⁢ 𝑑 y .$

Since $〈Iλ′⁢(un)-Iλ′⁢(u),un-u〉→0$ , we have

$( a + b ⁢ ∥ u n ∥ ( θ - 1 ) ⁢ p ) ⁢ [ u n , u n - u ] - ( a + b ⁢ ∥ u ∥ ( θ - 1 ) ⁢ p ) ⁢ [ u , u n - u ] = ∫ ℝ N ( | u n | p s * - 2 ⁢ u n - | u | p s * - 2 ⁢ u ) ⁢ ( u n - u ) ⁢ 𝑑 x + o ⁢ ( 1 ) .$

Thus,

$( a + b ⁢ ∥ u n ∥ ( θ - 1 ) ⁢ p ) ⁢ ( [ u n , u n - u ] - [ u , u n - u ] ) + ( a + b ⁢ ∥ u n ∥ ( θ - 1 ) ⁢ p ) ⁢ [ u , u n - u ] - ( a + b ⁢ ∥ u ∥ ( θ - 1 ) ⁢ p ) ⁢ [ u , u n - u ]$

$= ∫ ℝ N ( | u n | p s * - 2 ⁢ u n - | u | p s * - 2 ⁢ u ) ⁢ ( u n - u ) ⁢ 𝑑 x + o ⁢ ( 1 ) .$

By the boundedness of ${un}n$ and $un⇀u$ in $Ds,p⁢(ℝN)$ , we deduce

$lim n → ∞ ⁡ ( a + b ⁢ ∥ u n ∥ ( θ - 1 ) ⁢ p ) ⁢ [ u , u n - u ] = 0$

and

$lim n → ∞ ⁡ ( a + b ⁢ ∥ u ∥ ( θ - 1 ) ⁢ p ) ⁢ [ u , u n - u ] = 0 .$

Hence, we conclude from (4.28) that

$lim n → ∞ ⁡ ( a + b ⁢ ∥ u n ∥ ( θ - 1 ) ⁢ p ) ⁢ ( [ u n , u n - u ] - [ u , u n - u ] ) = 0 .$

This together with $d:=infn≥1⁡∥un∥>0$ and $b>0$ implies that

$lim n → ∞ ⁡ ( [ u n , u n - u ] - [ u , u n - u ] ) = 0 .$

Let us now recall the well-known Simon inequalities:

(4.29)

$| ξ - η | p ≤ { C p ′ ⁢ ( | ξ | p - 2 ⁢ ξ - | η | p - 2 ⁢ η ) ⋅ ( ξ - η ) for ⁢ p ≥ 2 , C p ′′ ⁢ [ ( | ξ | p - 2 ⁢ ξ - | η | p - 2 ⁢ η ) ⋅ ( ξ - η ) ] p / 2 ⁢ ( | ξ | p + | η | p ) ( 2 - p ) / 2 for ⁢ 1 < p < 2$

for all $ξ,η∈ℝN$ , where $Cp′$ and $Cp′′$ are positive constants depending only on p.

If $p>2$ , then it follows from (4.29) that

$∬ ℝ 2 ⁢ N | u n ⁢ ( x ) - u n ⁢ ( y ) - u ⁢ ( x ) + u ⁢ ( y ) | p | x - y | N + p ⁢ s ⁢ 𝑑 x ⁢ 𝑑 y ≤ C p ′ ⁢ ( [ u n , u n - u ] - [ u , u n - u ] ) → 0$

as $n→∞$ . Hence $un→u$ in $Ds,p⁢(ℝN)$ .

It remains to consider the case $1<p<2$ . To this end, from (4.29) we have

$∬ ℝ 2 ⁢ N | u n ⁢ ( x ) - u n ⁢ ( y ) - u ⁢ ( x ) + u ⁢ ( y ) | p | x - y | N + p ⁢ s ⁢ 𝑑 x ⁢ 𝑑 y$

$≤ C p ′′ ⁢ ( [ u n , u n - u ] - [ u , u n - u ] ) p / 2 ⁢ { ∬ ℝ 2 ⁢ N ( | u n ⁢ ( x ) - u n ⁢ ( y ) | + | u ⁢ ( x ) - u ⁢ ( y ) | ) p | x - y | N + p ⁢ s ⁢ 𝑑 x ⁢ 𝑑 y } ( 2 - p ) / 2$

$≤ C ⁢ ( [ u n , u n - u ] - [ u , u n - u ] ) p / 2 → 0$

as $n→∞$ . Hence $un→u$ in $Ds,p⁢(ℝN)$ . In conclusion, we get $un→u$ strongly in $Ds,p⁢(ℝN)$ as $n→∞$ .

Finally, we consider $infn⁡∥un∥Ds,p⁢(ℝN)=0$ . Then either 0 is an accumulation point of the sequence ${un}n$ and so there exists a subsequence of ${un}n$ strongly converging to $u=0$ , or 0 is an isolated point of the sequence ${un}n$ and so there exists a subsequence, still denoted by ${un}n$ , such that $infn⁡∥un∥>0$ . In the first case we are done, while in the latter case we can process as above. ∎

In the following, we set $Bρ⁢(0):={u∈Ds,p⁢(ℝN):∥u∥<ρ}$ .

Theorem 4.6.

There exists a constant $λ*>0$ such that for all $λ∈(0,λ*)$ problem (1.1) has a weak solution $u1$ in $Ds,p⁢(RN)$ with $Iλ⁢(u1)<0$ .

Proof.

By Lemma 4.1, there exists $λ0>0$ and $γ>0$ such that

$I λ ⁢ ( u ) ≥ γ > 0 for ⁢ λ ∈ ( 0 , λ 0 ) ⁢ and ⁢ u ∈ D s , p ⁢ ( ℝ N ) ⁢ with ⁢ ∥ u ∥ = ρ .$

Using Ekeland’s variational principle (see [16]) for the complete metric space $B¯ρ⁢(0):={u∈Ds,p⁢(ℝN):∥u∥≤ρ}$ with $d⁢(u,v)=∥u-v∥$ , we can prove that there exists a $(PS)c0$ sequence ${un}n⊂Bρ⁢(0)$ with

$c 0 = inf ⁡ { I λ ⁢ ( u ) : u ∈ B ¯ ρ ⁢ ( 0 ) } .$

Now we claim that $-∞<c0<0$ . Indeed, we can choose $φ0∈Ds,p⁢(ℝN)$ such that $∫ℝNf⁢φ0⁢𝑑x>0$ and fix $λ∈(0,λ*)$ . Then for any $t>0$ we have

$I λ ⁢ ( t ⁢ φ 0 ) = a ⁢ t p p ⁢ ∥ φ 0 ∥ p + b ⁢ t θ ⁢ p θ ⁢ p ⁢ ∥ φ 0 ∥ θ ⁢ p - t p s * p s * ⁢ ∫ ℝ N ( φ 0 + ) p s * ⁢ 𝑑 x - λ ⁢ t ⁢ ∫ ℝ N f ⁢ φ 0 ⁢ 𝑑 x ,$

which implies that there exists $t>0$ small enough such that $∥t⁢φ0∥≤ρ$ and $Iλ⁢(t⁢φ0)<0$ . Hence, $c0<I⁢(t⁢φ0)<0$ . It is easy to see that $c0>-∞$ .

Taking $λ*∈(0,λ0]$ such that

$0 < Λ - Θ ⁢ ( λ )$

for $λ∈(0,λ*)$ . We deduce from $c0<0$ and Lemma 4.5 that there exist a subsequence of ${un}n$ still denoted by ${un}n$ and $u1∈Ds,p⁢(ℝN)$ such that $un→u1$ strongly in $Ds,p⁢(ℝN)$ . Therefore, $Iλ′⁢(u1)=0$ and $Iλ⁢(u1)=c0<0$ , which ends the proof. ∎

Next we shall use arguments similar to those explored by Chabrowski [14], Alves [1] and Goncalves and Alves [20] to obtain that equation (1.1) has anther solution.

By Lemmas 4.1 and 4.2, we know that $Iλ$ satisfies the mountain pass geometry. Hence by the mountain pass theorem (see [3]), there exists a $(PS)c$ sequence ${vn}n$ with

$c = inf γ ∈ Γ ⁡ max t ∈ [ 0 , 1 ] ⁡ I λ ⁢ ( γ ⁢ ( t ) ) ,$

where

$Γ = { γ ∈ C ⁢ ( [ 0 , 1 ] , D s , p ⁢ ( ℝ N ) ) : γ ⁢ ( 0 ) = 0 , γ ⁢ ( 1 ) = e } .$

Lemma 4.7.

Assume $θ<N/(N-s⁢p)$ , $a≥0$ and $b>0$ , or $θ=N/(N-s⁢p)$ , $a>0$ and $0<b<S-θ$ . Then there exists $λ**∈(0,λ*)$ such that for any $λ∈(0,λ**)$ the inequality

$c ≤ sup t ≥ 0 ⁡ I λ ⁢ ( t ⁢ U 1 ) < Λ - Θ ⁢ ( λ )$

holds.

Proof.

Using (1.5), we consider the functions

$g ⁢ ( t ) := I λ ⁢ ( t ⁢ U 1 )$

$= a ⁢ t p p ⁢ ∥ U 1 ∥ p + b ⁢ t θ ⁢ p θ ⁢ p ⁢ ∥ U 1 ∥ θ ⁢ p - t p s * p s * ⁢ ∫ ℝ N U 1 p s * ⁢ 𝑑 x - λ ⁢ t ⁢ ∫ ℝ N f ⁢ U 1 ⁢ 𝑑 x$

$= a ⁢ t p p ⁢ S N s ⁢ p + b ⁢ t θ ⁢ p θ ⁢ p ⁢ S N ⁢ θ s ⁢ p - t p s * p s * ⁢ S N s ⁢ p - λ ⁢ t ⁢ ∫ ℝ N f ⁢ U 1 ⁢ 𝑑 x$

and

$g ~ ⁢ ( t ) := a ⁢ t p p ⁢ S N s ⁢ p + b ⁢ t θ ⁢ p θ ⁢ p ⁢ S N ⁢ θ s ⁢ p - t p s * p s * ⁢ S N s ⁢ p$

for all $t≥0$ . There exists $t1>0$ such that $g~′⁢(t1)=0$ and $maxt≥0⁡g~⁢(t)=g~⁢(t1)=Λ>0$ , where $t1$ satisfies

$t 1 = ( b ⁢ S N ⁢ ( θ - 1 ) s ⁢ p ) 1 p s * - θ ⁢ p$

$if ⁢ θ < N N - s ⁢ p , a = 0 , b > 0 ,$

$t 1 θ ⁢ p - p = b ⁢ S N ⁢ ( θ - 1 ) s ⁢ p + b 2 ⁢ S 2 ⁢ N ⁢ ( θ - 1 ) s ⁢ p + 4 ⁢ a 2$

$if ⁢ θ = N - s ⁢ p / 2 N - s ⁢ p , a , b > 0 ,$

$t 1 = ( a 1 - b ⁢ S θ ) 1 / ( θ - 1 ) ⁢ p$

$if ⁢ θ = N N - s ⁢ p , a > 0 , 0 < b < S - θ .$

Choose $λ1∈(0,λ*]$ such that

$Λ - Θ ⁢ ( λ 1 ) > 0 .$

Note that

$lim t → 0 + ⁡ ( a p ⁢ S N s ⁢ p ⁢ t p + b θ ⁢ p ⁢ S N ⁢ θ s ⁢ p ⁢ t θ ⁢ p ) = 0 .$

Hence there exists $t2∈(0,t1)$ such that for any $λ∈(0,λ1)$ we have

$max 0 ≤ t ≤ t 2 ⁡ g ⁢ ( t ) ≤ max 0 ≤ t ≤ t 2 ⁡ ( a p ⁢ S N s ⁢ p ⁢ t p + b θ ⁢ p ⁢ S N ⁢ θ s ⁢ p ⁢ t θ ⁢ p ) ≤ Λ - Θ ⁢ ( λ 1 ) < Λ - Θ ⁢ ( λ ) .$

We choose $λ**∈(0,λ1]$ such that for any $λ∈(0,λ**)$ we get

$λ ⁢ t 2 ⁢ ∫ ℝ N f ⁢ U 1 ⁢ 𝑑 x > ( a ⁢ S p - a ⁢ S θ ⁢ p ) 1 - p ′ ⁢ ( 1 - 1 θ ⁢ p ) p ′ ⁢ λ p ′ ⁢ | f | ( p s * ) ′ p ′ if ⁢ a > 0 ,$

and

$λ ⁢ t 2 ⁢ ∫ ℝ N f ⁢ U 1 ⁢ 𝑑 x > ( 1 θ ⁢ p - 1 p s * ) 1 - ( p s * ) ′ ⁢ ( 1 - 1 θ ⁢ p ) ( p s * ) ′ ⁢ λ ( p s * ) ′ ⁢ | f | ( p s * ) ′ ( p s * ) ′ if ⁢ a = 0 .$

Then for any $λ∈(0,λ**)$ one has

$sup t ≥ t 2 ⁡ g ⁢ ( t ) ≤ sup t ≥ t 2 ⁡ g ~ ⁢ ( t ) - λ ⁢ t 2 ⁢ ∫ ℝ N f ⁢ U 1 ⁢ 𝑑 x$

$< g ~ ⁢ ( t 1 ) - Θ ⁢ ( λ )$

$= Λ - Θ ⁢ ( λ ) .$

Thus for any $λ∈(0,λ**)$ we obtain

$c ≤ sup t ≥ 0 ⁡ I λ ⁢ ( t ⁢ U 1 ) = sup t ≥ 0 ⁡ g ⁢ ( t ) < Λ - Θ ⁢ ( λ ) .$

Therefore, we complete the proof. ∎

Theorem 4.8.

There exists a constant $λ**∈(0,λ*)$ such that for all $λ∈(0,λ**)$ problem (1.1) has a nontrivial and nonnegative solution $u2$ in $Ds,p⁢(RN)$ with $Iλ⁢(u2)>0$ .

Proof.

By Lemma 4.5 and Lemma 4.7, there exist a subsequence of ${vn}n$ , still denoted by ${vn}n$ , and a function $u2∈Ds,p⁢(ℝN)$ such that $vn→u2$ strongly in $Ds,p⁢(ℝN)$ . Therefore,

$I λ ′ ⁢ ( u 2 ) = 0 and I λ ⁢ ( u 2 ) = c > 0 = I λ ⁢ ( 0 ) ,$

which means that $u2$ is a nontrivial and nonnegative solution of (1.1). ∎

Proof of Theorem 1.5.

The proof of Theorem 1.5 follows immediately by the combination of Theorem 4.6 and Theorem 4.8. ∎

Communicated by Yiming Long

Funding source: National Natural Science Foundation of China

Award Identifier / Grant number: 11501565

Award Identifier / Grant number: 11601515

Award Identifier / Grant number: 11601103

Funding source: Natural Science Foundation of Heilongjiang Province

Award Identifier / Grant number: A201306

Funding statement: The first author was supported by Fundamental Research Funds for the Central Universities (no. 3122015L014) and the National Natural Science Foundation of China (no. 11501565, no. 11601515). The second author was supported by the Natural Science Foundation of Heilongjiang Province of China (no. A201306), the Research Foundation of Heilongjiang Educational Committee (no. 12541667) and the Doctoral Research Foundation of Heilongjiang Institute of Technology (no. 2013BJ15). The third author was supported by the National Natural Science Foundation of China (no. 11601103).

Acknowledgements

The authors would like to thank the anonymous referees for useful comments and valuable suggestions which helped us in depth to improve and clarify the paper greatly.

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Received: 2016-06-20

Revised: 2016-10-03

Accepted: 2016-10-03

Published Online: 2016-11-16

Published in Print: 2017-07-01

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

DOI: doi.org/10.1515/ans-2016-6002

Keywords for this article

Critical Sobolev Exponent; Principle of Concentration Compactness

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