Sharp Trudinger–Moser Inequality and Ground State Solutions to Quasi-Linear Schrödinger Equations with Degenerate Potentials in ℝn

Lu Chen; Guozhen Lu; Maochun Zhu

doi:10.1515/ans-2021-2146

Article Open Access

Sharp Trudinger–Moser Inequality and Ground State Solutions to Quasi-Linear Schrödinger Equations with Degenerate Potentials in ℝⁿ

Lu Chen , Guozhen Lu and Maochun Zhu

Published/Copyright: October 15, 2021

Published by

Become an author with De Gruyter Brill

Submit Manuscript Author Information Explore this Subject

From the journal Advanced Nonlinear Studies Volume 21 Issue 4

Abstract

The main purpose of this paper is to establish the existence of ground-state solutions to a class of Schrödinger equations with critical exponential growth involving the nonnegative, possibly degenerate, potential V:

$- div ⁡ ( | ∇ ⁡ u | n - 2 ⁢ ∇ ⁡ u ) + V ⁢ ( x ) ⁢ | u | n - 2 ⁢ u = f ⁢ ( u ) .$

To this end, we first need to prove a sharp Trudinger–Moser inequality in $ℝn$ under the constraint

$∫ ℝ n ( | ∇ ⁡ u | n + V ⁢ ( x ) ⁢ | u | n ) ⁢ 𝑑 x ≤ 1 .$

This is proved without using the technique of blow-up analysis or symmetrization argument. As far as what has been studied in the literature, having a positive lower bound has become a standard assumption on the potential $V⁢(x)$ in dealing with the existence of solutions to the above Schrödinger equation. Since $V⁢(x)$ is allowed to vanish on an open set in $ℝn$ , the loss of a positive lower bound of the potential $V⁢(x)$ makes this problem become fairly nontrivial. Our method to prove the Trudinger–Moser inequality in $ℝ2$ (see [L. Chen, G. Lu and M. Zhu, A critical Trudinger–Moser inequality involving a degenerate potential and nonlinear Schrödinger equations, Sci. China Math. 64 2021, 7, 1391–1410]) does not apply to this higher-dimensional case $ℝn$ for $n≥3$ here. To obtain the existence of a ground state solution, we use a non-symmetric argument to exclude the possibilities of vanishing and dichotomy cases of the minimizing sequence in the Nehari manifold. This argument is much simpler than the one used in dimension two where we consider the nonlinear Schrödinger equation $-Δ⁢u+V⁢u=f⁢(u)$ with a degenerate potential V in $ℝ2$ .

Keywords: Sharp Trudinger–Moser Inequalities; Degenerate Potential; Ground State Solutions; Schrödinger Equations; Nehari Manifold

MSC 2010: 35J10; 35J92; 46E35; 35J91; 26D10

1 Introduction

The Trudinger inequality as a borderline case of the Sobolev imbedding was obtained by Trudinger [32] (see also Pohozhaev [28]). More precisely, it was proved that there exists $α>0$ such that

(1.1)

$sup ∥ ∇ ⁡ u ∥ n n ≤ 1 , u ∈ W 0 1 , n ⁢ ( Ω ) ⁡ ∫ Ω exp ⁡ ( α ⁢ | u | n n - 1 ) ⁢ 𝑑 x < ∞ ,$

where $Ω⊆ℝn$ is a domain of finite measure and $W01,p⁢(Ω)$ denotes the usual Sobolev space, i.e. the completion of $C0∞⁢(Ω)$ with the norm

$∥ u ∥ W 1 , p ⁢ ( Ω ) = ( ∫ Ω ( | ∇ ⁡ u | p + | u | p ) ⁢ 𝑑 x ) 1 p .$

Subsequently, Trudinger’s inequality was sharpened by Moser in [27] by showing that the largest α in (1.1) is $αn=n⁢wn-11/(n-1)$ , where $ωn-1$ is the surface measure of the unit sphere in $ℝn$ . Inequality (1.1) in the case of $α=αn$ is known as the Trudinger–Moser inequality.

The Trudinger–Moser inequality was extended to the entire space $ℝn$ (see the works [5, 1, 21]). It was proved in [21] that

$sup u ∈ W 1 , n ⁢ ( ℝ n ) ∫ ℝ n ( | u | n + | ∇ ⁡ u | n ) ⁢ 𝑑 x ≤ 1 ⁡ ∫ ℝ n Φ n ⁢ ( α ⁢ | u | n n - 1 ) ⁢ 𝑑 x < + ∞ if and only if α ≤ α n ,$

where $Φn⁢(t)=et-∑k=0n-2tkk!$ .

We note that all proofs given above use the Pólya–Szegö inequality and symmetrization argument. A symmetrization-free argument has been found in [17, 16] which works for the non-Euclidean setting such as the Heisenberg group, hyperbolic spaces and Riemannian manifolds (see [23, 18]). This symmetrization-free argument can also be applied to establish the concentration-compactness principle of Trudinger–Moser type (see [19, 6, 34]) and Lorentz spaces [24].

One can easily obtain the following strengthened version of the Trudinger–Moser inequality in $ℝn$ (see, e.g., [17] by using a symmetrization-free method): for any $τ>0$ , there holds

(1.2)

$sup u ∈ W 1 , n ⁢ ( ℝ n ) ∫ ℝ n ( τ ⁢ | u | n + | ∇ ⁡ u | n ) ⁢ 𝑑 x ≤ 1 ⁡ ∫ ℝ n Φ n ⁢ ( α ⁢ | u | n n - 1 ) ⁢ 𝑑 x < + ∞ if and only if α ≤ α n .$

One can immediately deduce (1.2) with τ replaced by some $V⁢(x)$ which has a positive constant lower bound. Therefore, the following critical Trudinger–Moser inequality with a nondegenerate potential V of a positive lower bound is immediate:

(1.3)

$sup u ∈ W 1 , n ⁢ ( ℝ n ) ∫ ℝ n ( V ⁢ | u | n + | ∇ ⁡ u | n ) ⁢ 𝑑 x ≤ 1 ⁡ ∫ ℝ n Φ n ⁢ ( α ⁢ | u | n n - 1 ) ⁢ 𝑑 x < + ∞ if and only if α ≤ α n .$

As far as what has been studied in the literature, having a positive lower bound has become a standard assumption on the potential $V⁢(x)$ in dealing with the existence of solutions to the Schrödinger equation with the potential V:

(1.4)

${ - div ⁡ ( | ∇ ⁡ u | n - 2 ⁢ ∇ ⁡ u ) + V ⁢ ( x ) ⁢ | u | n - 2 ⁢ u = f ⁢ ( u ) in ⁢ ℝ n , u ∈ W 1 , n ⁢ ( ℝ n ) .$

For instance, when $V⁢(x)$ is a coercive potential, that is,

$V ⁢ ( x ) ≥ V 0 > 0 , and additionally either ⁢ lim x → ∞ ⁡ V ⁢ ( x ) = + ∞ ⁢ or ⁢ 1 V ∈ L 1 ⁢ ( ℝ n ) ,$

the existence and multiplicity results of equation (1.4) can be found in, e.g., [15] and the references therein. Their proofs depend crucially on the compact embeddings given by the coercive potential. When $V⁢(x)$ is a positive constant potential, i.e. $V⁢(x)=V0>0$ , the natural space for a variational treatment of (1.4) is $W1,n⁢(ℝn)$ . It is well known that the embedding $W1,n⁢(ℝn)↪Ln⁢(ℝn)$ is continuous but not compact, even in the class of radial functions. For the existence of nontrivial solutions in this case, we refer to [3, 31, 13, 26] and the references therein.

In the literature, the following Rabinowitz-type potentials were introduced in [30]:

(1.5)

$0 < V 0 = inf x ∈ ℝ n ⁡ V ⁢ ( x ) < sup x ∈ ℝ n ⁡ V ⁢ ( x ) = lim | x | → ∞ ⁡ V ⁢ ( x ) = V ∞ < + ∞ ,$

which are often involved in the study of equations with the subcritical polynomial growth; see, e.g., [22, 30, 25, 12, 33], the book [4] and many references therein. In dimension two, when the nonlinearity has critical exponential growth and the operator $-Δ$ in (1.4) is replaced by $-ε2⁢Δ$ , the existence of the semiclassical state $uε$ was obtained by Alves and Figueiredo [2] if ε is small enough. The authors of this paper [7] showed that it is possible to obtain the existence of ground state solutions under the assumptions (1.5) and $ε=1$ .

In the recent work [8], the authors of this paper proved a sharp Trudinger–Moser inequality in $ℝ2$ under the less restrictive constraint

$sup u ∈ H 1 ⁢ ( ℝ 2 ) , ∫ ℝ 2 ( | ∇ ⁡ u | 2 + V ⁢ ( x ) ⁢ | u | 2 ) ⁢ 𝑑 x ≤ 1 ⁡ ∫ ℝ 2 ( e 4 ⁢ π ⁢ | u | 2 - 1 ) ⁢ 𝑑 x < ∞ ,$

where $V⁢(x)$ is allowed to vanish on some open set of $ℝ2$ . We call this kind of potential function degenerate potential. The loss of a positive lower bound of the potential $V⁢(x)$ makes this problem become fairly complicated, since the classical methods, such as symmetrization and blow-up analysis [20, 21], fail in dealing with this problem. To overcome these difficulties, the authors applied the symmetrization-free argument developed in [17] and a new embedding theorem involving the degenerate potential. Furthermore, based on these inequalities and the Nehari manifold technique, they obtained the existence of the ground-state solution to the Schrödinger equation (1.4) involving the degenerate potential in dimension two.

More recently, the authors [10] established the existence of a least energy solution to the Q-sub-Laplacian equations with a constant or degenerate potential V on the Heisenberg group $ℍ=ℍn$ :

(1.6)

$- div ℍ ⁢ ( | ∇ ℍ ⁡ u | Q - 2 ⁢ ∇ ℍ ⁡ u ) + V ⁢ ( ξ ) ⁢ | u | Q - 2 ⁢ u = f ⁢ ( u )$

with the nonlinear term f of maximal exponential growth $exp⁡(α⁢tQ/(Q-1))$ as $t→+∞$ , where $ℍn=ℂn×ℝ$ is the n-dimensional Heisenberg group and $Q=2⁢n+2$ is the homogeneous dimension of $ℍn$ .

The authors established in [10] a sharp critical Trudinger–Moser inequality involving a degenerate potential on $ℍn$ . Then the existence of a ground state solution to the above equation (1.6) was proved when the potential is a positive constant or vanishes on some open bounded set of $ℍn$ .

Motivated by [8, 10] and our recent work on the existence of a least energy solution to a class of nonlinear biharmonic Schrödinger equations with constant and degenerate potentials in $ℝ4$ (see [9]), we will consider the quasilinear equations (1.4) when the potential V is neither a coercive potential nor a Rabinowitz-type potential. Namely, we will establish the existence of a ground state solution to equation (1.4) when V is degenerate on an open bounded set in $ℝn$ .

The first main result of this paper is to establish the following Trudinger–Moser inequality in higher dimension $ℝn$ under the constraint

$∫ ℝ n ( | ∇ ⁡ u | n + V ⁢ ( x ) ⁢ | u | n ) ⁢ 𝑑 x ≤ 1 ,$

where $V⁢(x)≥0$ satisfies

(V1)

$V ⁢ ( x ) = 0 ⁢ at ⁢ B δ ⁢ ( 0 ) ⁢ and ⁢ V ⁢ ( x ) ≥ V 0 ⁢ in ⁢ ℝ n ∖ B 2 ⁢ δ ⁢ ( 0 ) ⁢ for any ⁢ V 0 , δ > 0 .$

Theorem 1.1.

Assume that the potential $V⁢(x)$ satisfies condition (V1). Then

(1.7)

$sup u ∈ W 1 , n ⁢ ( ℝ n ) , ∫ ℝ n ( | ∇ ⁡ u | n + V ⁢ ( x ) ⁢ | u | n ) ⁢ 𝑑 x ≤ 1 ⁡ ∫ ℝ n Φ n ⁢ ( α n ⁢ | u | n n - 1 ) ⁢ 𝑑 x < ∞ .$

Our method of the proof is based on a new imbedding theorem involving degenerate (see Lemma 2.1) and a symmetrization-free method. This is inspired by our recent work on the Trudinger–Moser inequality with degenerate potentials on the entire Heisenberg group where the Pólya–Szegö inequality fails (see [10]). Next, we consider the existence of the ground-state solution of the Schrödinger equation (1.4) when the potential $V⁢(x)$ satisfies (V1) and

(V2)

$sup x ∈ ℝ n ⁡ V ⁢ ( x ) = lim | x | → ∞ ⁡ V ⁢ ( x ) = V ∞ > 0 ,$

and the nonlinear term $f⁢(t)∈C1$ satisfies the following conditions:

Critical exponential growth: there exists $α0>0$ such that

$lim | t | → + ∞ ⁡ | f ⁢ ( t ) | exp ⁡ ( α ⁢ | t | n n - 1 ) = { 0 for ⁢ α > α 0 , + ∞ for ⁢ α < α 0 .$
Ambrosetti–Rabinowitz (A-R) condition (see [29]): there exists $μ>n$ such that

$0 < μ ⁢ F ⁢ ( t ) := μ ⁢ ∫ 0 t f ⁢ ( s ) ⁢ 𝑑 s ≤ t ⁢ f ⁢ ( t ) for any ⁢ t ∈ ℝ .$
There exist $t0$ and $M0>0$ such that $F⁢(t)≤M0⁢|f⁢(t)|$ for any $|t|≥t0$ .
$f ⁢ ( 0 ) = 0$ and $f⁢(t)=o⁢(tn-1)$ as $t→0$ .
$f ⁢ ( t ) | t | n - 1$ is increasing.

Remark 1.2.

Condition (ii) implies that $F⁢(t)=o⁢(|t|n)$ as $t→0$ . Conditions (i), (ii) and (iv) imply the following growth condition: for any $ε>0$ and $β0>α0$ , there exists $Cε$ such that

(1.8)

$| f ⁢ ( t ) | ≤ ε ⁢ | t | n n - 1 + C ε ⁢ | t | μ - 1 ⁢ Φ n ⁢ ( β 0 ⁢ | t | n n - 1 ) for all ⁢ t ∈ ℝ .$

By condition (v), we know that the function $f⁢(t)⁢t-n⁢F⁢(t)$ is increasing.

Our second main result of this paper is the following theorem.

Theorem 1.3.

Under the assumptions (V1), (V2) and (i)–(v), if we further assume that $0<V∞<CT⁢M*⁢(F)$ , where

$C TM * ⁢ ( F ) := sup ⁡ { n ∥ u ∥ n n ⁢ ∫ ℝ n F ⁢ ( u ) ⁢ 𝑑 x | u ∈ W 1 , n ⁢ ( ℝ n ) ∖ { 0 } , ∥ ∇ ⁡ u ∥ n ≤ ( α n α 0 ) n - 1 n } ,$

then equation (1.4) admits a positive ground state solution.

The Trudinger–Moser ratio $CT⁢M*⁢(F)$ was introduced in [14] in dimension 2. The authors of [26] showed in higher dimensions that $CT⁢M*⁢(F)=+∞$ is equivalent to (see [26, Theorem 7.4])

$lim t → + ∞ ⁡ F ⁢ ( t ) ⁢ | t | n n - 1 exp ⁡ ( α 0 ⁢ | t | n n - 1 ) = + ∞ .$

Hence, we immediately have the following corollary.

Corollary 1.4.

Under assumptions (V1), (V2) and (i)–(v), if we further assume that

$lim t → + ∞ ⁡ F ⁢ ( t ) ⁢ | t | n n - 1 exp ⁡ ( α 0 ⁢ | t | n n - 1 ) = + ∞ ,$

then equation (1.4) admits a positive ground state solution.

We will prove this theorem by the Nehari manifold technique similar to that used in $ℝ2$ in [8]. However, unlike in [8], we will directly exclude the possibilities of vanishing and dichotomy cases of the minimizing sequence in the Nehari manifold through a simpler non-symmetric argument; for the details see Lemmas 3.2 and 3.3. We should note that this string of similar ideas has been used in our recent work in the subelliptic setting [10].

We have also recently established a sharp Trudinger-Moser inequality on an n-dimensional complete and noncompact Riemannian manifold $(M,g)$ under the constraint

$∫ M ( | ∇ ⁡ u | n + V ⁢ ( ξ ) ⁢ | u | n ) ⁢ 𝑑 V g ≤ 1 ,$

where $V⁢(ξ)≥0$ satisfies

(V3)

$V ⁢ ( x ) = 0 ⁢ at ⁢ B r ⁢ ( ξ 0 ) ⁢ and ⁢ V ⁢ ( ξ ) ≥ C 0 ⁢ in ⁢ M ∖ B 2 ⁢ δ ⁢ ( ξ 0 ) for some ⁢ ξ 0 ∈ M , C 0 , r > 0 .$

More precisely, we have proved in [11] the following theorem.

Theorem 1.5.

Let $(M,g)$ be an n-dimensional complete noncompact manifold $(M,g)$ with its Ricci curvature having the lower bound, namely $R⁢c⁢(M,g)≥λ⁢g$ for some constant $λ∈R$ , and its injectivity radius being $inj⁡(M,g)≥i0>0$ . Assume that the potential $V⁢(ξ)$ satisfies condition (V3). Then

$sup u ∈ W 1 , n ⁢ ( M ) , ∫ M ( | ∇ ⁡ u | n + V ⁢ ( ξ ) ⁢ | u | n ) ⁢ 𝑑 V g ≤ 1 ⁡ ∫ M Φ n ⁢ ( α n ⁢ | u | n n - 1 ) ⁢ 𝑑 V g < ∞ .$

We remark that when V is a positive constant, the critical Trudinger–Moser inequality on complete and noncompact Riemannian manifolds was established in [18].

As an application of Theorem 1.5, we give necessary conditions for the existence of the ground-state solution of the following quasi-linear Schrödinger equation:

$- div ⁡ ( | ∇ g ⁡ u | n - 2 ⁢ ∇ g ⁡ u ) + V ⁢ ( ξ ) ⁢ | u | n - 2 ⁢ u = f ⁢ ( u ) in ⁢ M ,$

where $V⁢(ξ)$ is allowed to vanish on an open and bounded domain of M and $f⁢(t)$ satisfies a certain critical exponential growth.

Throughout this paper, the letter c always denotes some positive constant which may vary from line to line.

The organization of this paper is as follows. Section 2 is devoted to proving Theorem 1.1, namely the Trudinger–Moser inequality under the constraint $∫ℝn(|∇⁡u|n+V⁢(x)⁢|u|n)⁢𝑑x≤1$ , where $V⁢(x)$ vanishes on some open set of $ℝn$ . Section 3 gives the proof of Theorem 1.3, namely the existence of the ground state solution to the nonlinear Schrödinger equation (1.4) involving the degenerate potential V.

2 The Proof of Theorem 1.1

In this section, we will prove the sharp Trudinger–Moser inequality involving the degenerate potential, namely we shall give the proof of Theorem 1.1. We need the following new imbedding theorem whose proof in dimension two [8] cannot be adapted in higher dimension here. Instead, we follow the idea from [10].

Lemma 2.1.

Assume that $u∈W1,n⁢(Rn)$ such that

$∫ ℝ n ( | ∇ ⁡ u | n + V ⁢ ( x ) ⁢ | u | n ) ⁢ 𝑑 x < + ∞ ,$

where $V⁢(x)$ satisfies assumption (V1). Then there exists some constant $c>0$ depending on the dimension n, δ and $V0$ in (V1) such that

$∫ ℝ n | u | n ⁢ 𝑑 x ≤ c ⁢ ∫ ℝ n ( | ∇ ⁡ u | n + V ⁢ ( x ) ⁢ | u | n ) ⁢ 𝑑 x .$

Proof.

Choose a smooth function φ such that $φ=1$ in $B2⁢δ$ , and $φ=0$ in $ℝn∖B4⁢δ$ , and $0≤φ≤1,|∇⁡φ|≤cδ$ . By the Sobolev–Poincaré inequality, we have

$∫ B 4 ⁢ δ | u ⁢ φ | n ⁢ 𝑑 x ≤ c ⁢ δ n ⁢ ∫ B 4 ⁢ δ | ∇ ⁡ ( u ⁢ φ ) | n ⁢ 𝑑 x$

$≤ c ⁢ δ n ⁢ ∫ B 4 ⁢ δ | φ ⁢ ∇ ⁡ u | n ⁢ 𝑑 x + c ⁢ ∫ B 4 ⁢ δ ∖ B 2 ⁢ δ | u | n ⁢ 𝑑 x + ∑ k = 1 n - 1 c ⁢ δ n ⁢ ∫ B 4 ⁢ δ | ( φ ⁢ ∇ ⁡ u ) k ⁢ ( u ⁢ ∇ ⁡ φ ) n - k | ⁢ 𝑑 x$

$≤ c ⁢ δ n ⁢ ∫ B 4 ⁢ δ | φ ⁢ ∇ ⁡ u | n ⁢ 𝑑 x + c ⁢ ∫ B 4 ⁢ δ ∖ B 2 ⁢ δ | u | n ⁢ 𝑑 x + ∑ k = 1 n - 1 c ⁢ δ n ⁢ ∫ B 4 ⁢ δ ∖ B 2 ⁢ δ | ( ∇ ⁡ u ) k ⁢ u n - k | ⁢ 𝑑 x ,$

where c is a positive constant depending only on the dimension n. Applying Hölder’s inequality with the conjugate exponents $p=nk$ and $q=nn-k$ to estimate the integral

$∫ B 4 ⁢ δ ∖ B 2 ⁢ δ | ( ∇ ⁡ u ) k ⁢ u n - k | ⁢ 𝑑 x ,$

we get

$∫ B 2 ⁢ δ | u | n ⁢ 𝑑 x ≤ ∫ B 4 ⁢ δ | u ⁢ φ | n ⁢ 𝑑 x$

$≤ c ⁢ δ n ⁢ ∫ B 4 ⁢ δ | ∇ ⁡ u | n ⁢ 𝑑 x + c ⁢ ∫ B 4 ⁢ δ ∖ B 2 ⁢ δ | u | n ⁢ 𝑑 x + ∑ k = 1 n - 1 c ⁢ δ n ⁢ ∫ B 4 ⁢ δ ∖ B 2 ⁢ δ | ( ∇ ⁡ u ) k ⁢ u n - k | ⁢ 𝑑 x$

$≤ c ⁢ δ n ⁢ ∫ B 4 ⁢ δ | ∇ ⁡ u | n ⁢ 𝑑 x + c ⁢ ∫ B 4 ⁢ δ ∖ B 2 ⁢ δ | u | n ⁢ 𝑑 x + ∑ k = 1 n - 1 c ⁢ δ n ⁢ ∫ B 4 ⁢ δ ∖ B 2 ⁢ δ ( | ∇ ⁡ u | n + | u | n ) ⁢ 𝑑 x$

$≤ c 1 ⁢ ∫ B 4 ⁢ δ | ∇ ⁡ u | n ⁢ 𝑑 x + c 2 ⁢ ∫ B 4 ⁢ δ ∖ B 2 ⁢ δ | u | n ⁢ 𝑑 x ,$

where $c1,c2$ depend on n and δ. Recall that $V⁢(x)≥V0$ in $B4⁢δc$ . Then

$∫ B 2 ⁢ δ | u | n ⁢ 𝑑 x + ∫ B 4 ⁢ δ c | u | n ⁢ 𝑑 x ≤ c 1 ⁢ ∫ B 4 ⁢ δ | ∇ ⁡ u | n ⁢ 𝑑 x + c 2 ⁢ ∫ B 4 ⁢ δ ∖ B 2 ⁢ δ | u | n ⁢ 𝑑 x + 1 V 0 ⁢ ∫ B 4 ⁢ δ c V ⁢ ( x ) ⁢ | u | n ⁢ 𝑑 x ,$

and thus

$∫ B 2 ⁢ δ | u | n ⁢ 𝑑 x + ∫ B 4 ⁢ δ c | u | n ⁢ 𝑑 x + ∫ B 4 ⁢ δ ∖ B 2 ⁢ δ | u | n ⁢ 𝑑 x ≤ c 1 ⁢ ∫ B 4 ⁢ δ | ∇ ⁡ u | n ⁢ 𝑑 x + ( c 2 + 1 ) ⁢ ∫ B 4 ⁢ δ ∖ B 2 ⁢ δ | u | n ⁢ 𝑑 x + 1 V 0 ⁢ ∫ B 4 ⁢ δ c V ⁢ ( x ) ⁢ | u | n ⁢ 𝑑 x .$

Recall again that $V⁢(x)≥V0$ in $B2⁢δc$ . Then we get

$∫ ℝ n | u | n ⁢ 𝑑 x ≤ c 1 ⁢ ∫ ℝ n | ∇ ⁡ u | n ⁢ 𝑑 x + c 2 + 1 + V 0 V 0 ⁢ ∫ ℝ n V ⁢ ( x ) ⁢ | u | n ⁢ 𝑑 x .$

Taking $c=max⁡{c1,V0+1+c2V0}$ , we conclude the proof of Lemma 2.1. ∎

Remark 2.2.

If we define $WV1,n⁢(ℝn)$ as the completion of $Cc∞⁢(ℝn)$ under the norm

$( ∫ ℝ n | ∇ ⁡ u | n + V ⁢ ( x ) ⁢ | u | n ⁢ d ⁢ x ) 1 n ,$

then Lemma 2.1 implies that $WV1,n⁢(ℝn)=W1,n⁢(ℝn)$ .

Now, we are in the position to prove Theorem 1.1. This is inspired by the method used on the Heisenberg group [10].

Proof of Theorem 1.1.

Without loss of generality, we can assume that $u≥0$ is a compactly supported smooth function. Furthermore, we assume that $∫ℝnV⁢(x)⁢|u|n⁢𝑑x>0$ . Indeed, if $∫ℝnV⁢(x)⁢|u|n⁢𝑑x=0$ , by the assumptions on $V⁢(x)$ , we get $supp⁡u⊆B2⁢δ⁢(0)$ . Then, using the classical Trudinger–Moser inequality on bounded domains, we have

$∫ ℝ n Φ n ⁢ ( α n ⁢ | u | n n - 1 ) ⁢ 𝑑 x = ∫ B 2 ⁢ δ exp ⁡ ( α n ⁢ | u | n n - 1 ) ⁢ 𝑑 x < c ,$

where the constant c is independent of u, and the proof of Theorem 1.1 is completed. Hence, it remains to consider the case when $∫ℝnV⁢(x)⁢|u|n⁢𝑑x>0$ .

Set

$A ⁢ ( u ) := 2 - 1 n ⁢ ( n - 1 ) ⁢ ( ∫ ℝ n V ⁢ ( x ) ⁢ | u | n ⁢ 𝑑 x ) 1 n$

and $Ω(u):={u>A(u)}$ . Then $A⁢(u)<1$ and

$∫ Ω ⁢ ( u ) ∩ B 2 ⁢ δ c | u | n ⁢ 𝑑 x ≥ ∫ Ω ⁢ ( u ) ∩ B 2 ⁢ δ c | A ⁢ ( u ) | n ⁢ 𝑑 x = 2 - 1 n - 1 ⁢ ( ∫ ℝ n V ⁢ ( x ) ⁢ | u | n ⁢ 𝑑 x ) ⁢ | Ω ⁢ ( u ) ∩ B 2 ⁢ δ c | .$

Hence by (V1), we get

$| Ω ⁢ ( u ) ∩ B 2 ⁢ δ c | ⩽ 2 1 n - 1 ⁢ ∫ Ω ⁢ ( u ) ∖ B 2 ⁢ δ | u | n ⁢ 𝑑 x ∫ ℝ n V ⁢ ( x ) ⁢ | u | n ⁢ 𝑑 x ≤ 2 1 n - 1 V 0 ,$

and thus $|Ω⁢(u)|≤c⁢(δ,V0)$ .

Now, we rewrite

$∫ ℝ n Φ n ( α n | u | n n - 1 ) d x = ∫ Ω ⁢ ( u ) Φ n ( α n | u | n n - 1 ) d x + ∫ ℝ n ∖ Ω ⁢ ( u ) Φ n ( α n | u | n n - 1 ) d x = : I 1 + I 2 ,$

and we will show that both $I1$ and $I2$ are bounded.

First, we estimate $I2$ . Since

$∫ ℝ n ( | ∇ ⁡ u | n + V ⁢ ( x ) ⁢ | u | n ) ⁢ 𝑑 x ≤ 1 ,$

we know that $∫ℝn|u|n⁢𝑑x$ is bounded from Lemma 2.1, and thus

$I 2 ⩽ ∫ { u ( x ) < 1 } ∑ k = n - 1 ∞ | α n | k k ! ⁢ | u | k ⁢ n n - 1 ⁢ d ⁢ x ⩽ ∑ k = n - 1 ∞ | α n | k k ! ⁢ ∫ ℝ n | u | n ⁢ 𝑑 x ⩽ c .$

Now, we estimate $I1$ . Setting

$v ⁢ ( x ) = u ⁢ ( x ) - A ⁢ ( u ) in ⁢ Ω ⁢ ( u ) ,$

we obtain $v∈W01,n⁢(Ω⁢(u))$ . Moreover, by using Young’s inequality, we have

$| u ⁢ ( x ) | n n - 1 = | v ⁢ ( x ) + A ⁢ ( u ) | n n - 1$

$≤ | v ⁢ ( x ) | n n - 1 + n n - 1 ⁢ 2 1 n - 1 ⁢ ( | v | 1 n - 1 ⁢ A ⁢ ( u ) + | A ⁢ ( u ) | n n - 1 )$

$≤ | v ⁢ ( x ) | n n - 1 + n n - 1 ⁢ 2 1 n - 1 ⁢ | v | n n - 1 ⁢ | A ⁢ ( u ) | n n + 2 1 n - 1 + n n - 1 ⁢ 2 1 n - 1 ⁢ | A ⁢ ( u ) | n n - 1$

$= | v ⁢ ( x ) | n n - 1 ⁢ ( 1 + 2 1 n - 1 n - 1 ⁢ | A ⁢ ( u ) | n ) + 2 1 n - 1 + n n - 1 ⁢ 2 1 n - 1 ⁢ | A ⁢ ( u ) | n n - 1 ,$

where we have used the following elementary inequality:

$( a + b ) q ⩽ a q + q ⁢ 2 q - 1 ⁢ ( a q - 1 ⁢ b + b q ) for all ⁢ q ⩾ 1 ⁢ and ⁢ a , b ⩾ 0 .$

Let

$w ⁢ ( x ) = v ⁢ ( x ) ⁢ ( 1 + 2 1 n - 1 n - 1 ⁢ | A ⁢ ( u ) | n ) n - 1 n .$

Then clearly we have

$w ⁢ ( x ) ∈ W 0 1 , n ⁢ ( Ω ⁢ ( u ) ) ,$

$| u ⁢ ( x ) | n n - 1 ≤ | w ⁢ ( x ) | n n - 1 + 2 1 n - 1 + n n - 1 ⁢ 2 1 n - 1 ⁢ | A ⁢ ( u ) | n n - 1 ,$

$∫ Ω ⁢ ( u ) | ∇ ⁡ w ⁢ ( x ) | n ⁢ 𝑑 x = ( 1 + 2 1 n - 1 n - 1 ⁢ | A ⁢ ( u ) | n ) n - 1 ⁢ ∫ Ω ⁢ ( u ) | ∇ ⁡ v ⁢ ( x ) | n ⁢ 𝑑 x$

$≤ ( 1 + 2 1 n - 1 n - 1 ⁢ | A ⁢ ( u ) | n ) n - 1 ⁢ ( 1 - ∫ ℝ n V ⁢ ( x ) ⁢ | u | n ⁢ 𝑑 x ) .$

Thus, from the definition of $A⁢(u)$ we have

$( ∫ Ω ⁢ ( u ) | ∇ ⁡ w ⁢ ( x ) | n ⁢ 𝑑 x ) 1 n - 1 ≤ ( 1 + 2 1 n - 1 n - 1 ⁢ | A ⁢ ( u ) | n ) ⁢ ( 1 - ∫ ℝ n V ⁢ ( x ) ⁢ | u | n ⁢ 𝑑 x ) 1 n - 1$

$≤ ( 1 + 2 1 n - 1 n - 1 ⁢ | A ⁢ ( u ) | n ) ⁢ ( 1 - 1 n - 1 ⁢ ∫ ℝ n V ⁢ ( x ) ⁢ | u | n ⁢ 𝑑 x )$

$≤ ( 1 + 1 n - 1 ⁢ ∫ ℝ n V ⁢ ( x ) ⁢ | u | n ⁢ 𝑑 x ) ⁢ ( 1 - 1 n - 1 ⁢ ∫ ℝ n V ⁢ ( x ) ⁢ | u | n ⁢ 𝑑 x ) ,$

where we have used the following inequality:

$( 1 - x ) q ⩽ 1 - q ⁢ x for all ⁢ 0 ⩽ x ⩽ 1 , 0 < q ⩽ 1 .$

Then, using the classical Trudinger–Moser inequalities on bounded domains (see [27]), we get

$I 1 ≤ ∫ Ω ⁢ ( u ) Φ n ⁢ ( α n ⁢ | u | n n - 1 ) ⁢ 𝑑 x ≤ e α n ⁢ ( 2 1 n - 1 + n n - 1 ⁢ 2 1 n - 1 ⁢ | A ⁢ ( u ) | n n - 1 ) ⁢ ∫ Ω ⁢ ( u ) e α n ⁢ | w | n n - 1 ⁢ 𝑑 x ≤ c ,$

and the proof is finished. ∎

3 Existence of Ground State Solutions to Schrödinger Equations Involving the Degenerate Potential

In this section, we are concerned with the ground state solutions of the following nonlinear Schrödinger equation:

(3.1)

$- div ⁡ ( | ∇ ⁡ u | n - 2 ⁢ ∇ ⁡ u ) + V ⁢ ( x ) ⁢ | u | n - 2 ⁢ u = f ⁢ ( u ) ,$

where $V⁢(x)≥0$ satisfies (V1) and (V2), and f satisfies (i)–(v).

The associated functional is

$I V ⁢ ( u ) = 1 n ⁢ ∫ ℝ n ( | ∇ ⁡ u | n + V ⁢ ( x ) ⁢ | u | n ) ⁢ 𝑑 x - ∫ ℝ n F ⁢ ( u ) ⁢ 𝑑 x ,$

where $F⁢(t)=∫0tf⁢(s)⁢𝑑s$ , and its Nehari manifold is

$𝒩 V = { u ∈ W 1 , n ⁢ ( ℝ n ) ∣ u ≠ 0 , N V ⁢ ( u ) = 0 } ,$

where

$N V ⁢ ( u ) = ∫ ℝ n ( | ∇ ⁡ u | n + V ⁢ ( x ) ⁢ | u | n ) ⁢ 𝑑 x - ∫ ℝ n f ⁢ ( u ) ⁢ u ⁢ 𝑑 x .$

To study equation (3.1), we need the following limiting equation:

(3.2)

$- div ⁡ ( | ∇ ⁡ u | n - 2 ⁢ ∇ ⁡ u ) + V ∞ ⁢ | u | n - 2 ⁢ u = f ⁢ ( u ) ,$

where we recall from (V2) that $supx∈ℝn⁡V⁢(x)=lim|x|→∞⁡V⁢(x)=V∞>0$ .

The corresponding functional and Nehari manifold associated with (3.2) are

$I ∞ ⁢ ( u ) = 1 n ⁢ ∫ ℝ n ( | ∇ ⁡ u | n + V ∞ ⁢ | u | n ) ⁢ 𝑑 x - ∫ ℝ n F ⁢ ( u ) ⁢ 𝑑 x$

and

$𝒩 ∞ = { u ∈ W 1 , n ⁢ ( ℝ n ) ∣ u ≠ 0 , N ∞ ⁢ ( u ) = 0 } ,$

where

$N ∞ ⁢ ( u ) = ∫ ℝ n ( | ∇ ⁡ u | n + V ∞ ⁢ | u | n ) ⁢ 𝑑 x - ∫ ℝ n f ⁢ ( u ) ⁢ u ⁢ 𝑑 x .$

One can easily verify that if $u∈𝒩V$ or $u∈𝒩∞$ , then $IV$ and $I∞$ can be expressed by

$1 n ⁢ ∫ ℝ n ( f ⁢ ( u ) ⁢ u - n ⁢ F ⁢ ( u ) ) ⁢ 𝑑 x .$

Next, we follow the general framework and modify the method used in [10] on the Heisenberg group and apply it to the Euclidean spaces. For the convenience of the reader, we will show all details. Setting $mV=inf⁡{IV⁢(u)∣NV⁢(u)=0}$ , we will show that $mV$ can be attained by some function $u∈W1,n⁢(ℝn)$ . First, we give the following important observation for $mV$ .

Lemma 3.1.

There holds

(3.3)

$0 < m V < 1 n ⁢ ( α n α 0 ) n - 1 .$

Proof.

First, we prove $mV>0$ by contradiction. If there exists some sequence $uk$ in $𝒩V$ such that $IV⁢(uk)→0$ , that is,

(3.4)

$lim k → + ∞ ⁡ ∫ ℝ n ( f ⁢ ( u k ) ⁢ u k - n ⁢ F ⁢ ( u k ) ) ⁢ 𝑑 x = 0 ,$

without loss of generality, we can assume $uk≥0$ . From $uk∈𝒩V$ , (3.4) and the (A-R) condition, we can obtain

$lim k → + ∞ ⁡ ∫ ℝ n ( | ∇ ⁡ u k | n + V ⁢ ( x ) ⁢ u k n ) ⁢ 𝑑 x = 0 .$

Then, on one hand, it follows from (1.8) and $uk∈𝒩V$ that

(3.5)

$1 = ∫ ℝ n f ⁢ ( u k ) ⁢ u k ∥ u k ∥ W V 1 , n ⁢ ( ℝ n ) n ⁢ 𝑑 x ≤ ∫ ℝ n ( ε ⁢ u k n ∥ u k ∥ W V 1 , n ⁢ ( ℝ n ) n + C ε ⁢ u k μ ∥ u k ∥ W V 1 , n ⁢ ( ℝ n ) n ⁢ Φ n ⁢ ( β 0 ⁢ u k n n - 1 ) ) ⁢ 𝑑 x$

On the other hand, by the Trudinger–Moser inequality (1.7) and $μ>n$ , we get for any $p>1$ ,

$1 ∥ u k ∥ W V 1 , n ⁢ ( ℝ n ) n ⁢ ∫ ℝ n u k μ ⁢ Φ n ⁢ ( β 0 ⁢ u k n n - 1 ) ⁢ 𝑑 x ≤ 1 ∥ u k ∥ W V 1 , n ⁢ ( ℝ n ) n ⁢ ( ∫ ℝ n u k μ ⁢ p ⁢ 𝑑 x ) 1 / p ⁢ ( ∫ ℝ n Φ n ⁢ ( p ′ ⁢ β 0 ⁢ u k n n - 1 ) ⁢ 𝑑 x ) 1 / p ′$

$≤ c ⁢ ∥ u k ∥ W V 1 , n ⁢ ( ℝ n ) μ - n → 0 as ⁢ k → ∞ ,$

which is a contradiction to (3.5).

Next, we show that $mV<1n⁢(αnα0)n-1$ . The authors of [26] have proved that

$m ∞ = I ∞ ⁢ ( w ) < 1 n ⁢ ( α n α 0 ) n - 1$

and is achieved by some nonnegative function $w∈W1,n⁢(ℝn)$ under the hypothesis of Theorem 1.3. Since $w∈𝒩∞$ , we have

$∫ ℝ n ( | ∇ ⁡ w | n + V ∞ ⁢ w n ) ⁢ 𝑑 x = ∫ ℝ n f ⁢ ( w ) ⁢ w ⁢ 𝑑 x ,$

which implies that

$∫ ℝ n ( | ∇ ⁡ w | n + V ⁢ ( x ) ⁢ w n ) ⁢ 𝑑 x < ∫ ℝ n f ⁢ ( w ) ⁢ w ⁢ 𝑑 x .$

It follows that there exists $t∈(0,1)$ such that $t⁢w∈𝒩V$ and

$m V ≤ I V ⁢ ( t ⁢ w )$

$= 1 n ⁢ ∫ ℝ n ( f ⁢ ( t ⁢ w ) ⁢ t ⁢ w - n ⁢ F ⁢ ( t ⁢ w ) ) ⁢ 𝑑 x$

$≤ 1 n ⁢ ∫ ℝ n ( f ⁢ ( w ) ⁢ w - n ⁢ F ⁢ ( w ) ) ⁢ 𝑑 x$

$= I ∞ ⁢ ( w ) = m ∞ .$

This proves that

$m V ≤ m ∞ < 1 n ⁢ ( α n α 0 ) n - 1 ,$

as desired. ∎

Assume ${uk}k⊂𝒩V$ is a minimizing sequence for $mV$ . By a standard argument, one can easily derive that ${uk}k$ is bounded in $W1,n⁢(ℝn)$ . Hence, up to a subsequence, there exists $u∈W1,n⁢(ℝn)$ such that, for any $p>1$ , $uk→u$ weakly in $W1,n⁢(ℝn)$ and in $Lp⁢(ℝn)$ , and $uk→u$ in $Llocp⁢(ℝn)$ .

Now, we claim the following lemma.

Lemma 3.2.

One has $u≠0$ .

Proof.

We prove this by contradiction. If $u=0$ , then $uk→0$ in $Llocn⁢(ℝn)$ . By (V2) and a standard argument (see [8, Lemma 3.4]), we have

(3.6)

$lim k → + ∞ ⁡ ∫ ℝ n ( V ∞ - V ⁢ ( x ) ) ⁢ u k n ⁢ 𝑑 x = 0 .$

Since $uk∈𝒩V$ , we have

(3.7)

$∫ ℝ n ( | ∇ ⁡ u k | n + V ⁢ ( x ) ⁢ u k n ) ⁢ 𝑑 x - ∫ ℝ n f ⁢ ( u k ) ⁢ u k ⁢ 𝑑 x = 0 ,$

and there exists some sequence $tk≥1$ such that $tk⁢uk∈𝒩∞$ , that is,

(3.8)

$∫ ℝ n ( | ∇ ⁡ u k | n + V ∞ ⁢ u k n ) ⁢ 𝑑 x - ∫ ℝ n f ⁢ ( t k ⁢ u k ) ⁢ u k t k n - 1 ⁢ 𝑑 x = 0 .$

Combining (3.7) and (3.8), we get

$∫ ℝ n f ⁢ ( t k ⁢ u k ) ( t k ⁢ u k ) n - 1 ⁢ u k n ⁢ 𝑑 x - ∫ ℝ n f ⁢ ( u k ) u k n - 1 ⁢ u k n ⁢ 𝑑 x = ∫ ℝ n ( V ∞ - V ⁢ ( x ) ) ⁢ u k n ⁢ 𝑑 x → 0 .$

Therefore, we have

(3.9)

$∫ ℝ n f ⁢ ( t k ⁢ u k ) ( t k ⁢ u k ) n - 1 ⁢ u k n ⁢ 𝑑 x = ∫ ℝ n f ⁢ ( u k ) u k n - 1 ⁢ u k n ⁢ 𝑑 x + o k ⁢ ( 1 ) .$

Next, we show that $tk→t0=1$ by contradiction. Assume that $t0>1$ . Since

$I V ⁢ ( u k ) = 1 n ⁢ ∫ ℝ n ( f ⁢ ( u k ) ⁢ u k - n ⁢ F ⁢ ( u k ) ) ⁢ 𝑑 x → m V > 0 as ⁢ k → ∞ ,$

we have

(3.10)

$lim k → ∞ ⁡ ∫ ℝ n f ⁢ ( u k ) ⁢ u k ⁢ 𝑑 x > 0 .$

Now, we claim that

(3.11)

$lim k → ∞ ⁡ ∫ ℝ n f ⁢ ( t 0 ⁢ u k ) ( t 0 ⁢ u k ) n - 1 ⁢ u k n ⁢ 𝑑 x > lim k → ∞ ⁡ ∫ ℝ n f ⁢ ( u k ) ⁢ u k ⁢ 𝑑 x .$

Since $∥uk∥n$ is bounded and $f⁢(t)=o⁢(tn-1)$ when t is small, we obtain

$lim r → 0 ⁡ lim k → + ∞ ⁡ ∫ { u k < r } f ⁢ ( u k ) ⁢ u k ⁢ 𝑑 x = 0 .$

Hence, from (3.10), one of the following two cases must happen.

Case 1. It holds

$lim R → + ∞ ⁡ lim k → ∞ ⁡ ∫ { u k > R } f ⁢ ( u k ) ⁢ u k ⁢ 𝑑 x > 0 .$

Case 2. There exist some $r,R>0$ such that

$lim k → ∞ ⁡ ∫ { r < u k < R } f ⁢ ( u k ) ⁢ u k ⁢ 𝑑 x > 0 .$

If case 1 occurs, because $f⁢(t)$ has critical exponential growth when t is large enough, then we have

$lim R → + ∞ ⁡ lim k → ∞ ⁡ ∫ { u k > R } f ⁢ ( t 0 ⁢ u k ) ( t 0 ⁢ u k ) n - 1 ⁢ u k n ⁢ 𝑑 x > lim R → + ∞ ⁡ lim k → ∞ ⁡ ∫ { u k > R } f ⁢ ( u k ) ⁢ u k ⁢ 𝑑 x .$

If case 2 occurs, since the $uk$ are bounded on the set ${x∣r≤uk≤R}$ , then we have $|{x∣r≤uk≤R}|>β$ for some $β>0$ . Hence,

$lim k → ∞ ⁡ ∫ { r ≤ u k ≤ R } f ⁢ ( t 0 ⁢ u k ) ( t 0 ⁢ u k ) n - 1 ⁢ u k n ⁢ 𝑑 x > lim k → ∞ ⁡ ∫ { r ≤ u k ≤ R } f ⁢ ( u k ) ⁢ u k ⁢ 𝑑 x .$

By combining the above estimates and the monotonicity of $f⁢(t)tn-1$ , the claim (3.11) is proved. Therefore, we derive that

$lim k → ∞ ⁡ ∫ ℝ n f ⁢ ( t k ⁢ u k ) ( t k ⁢ u k ) n - 1 ⁢ u k n ⁢ 𝑑 x ≥ lim k → ∞ ⁡ ∫ ℝ n f ⁢ ( t 0 ⁢ u k ) ( t 0 ⁢ u k ) n - 1 ⁢ u k n ⁢ 𝑑 x > lim k → ∞ ⁡ ∫ ℝ n f ⁢ ( u k ) ⁢ u k ⁢ 𝑑 x > 0 ,$

which contradicts (3.9), and thus $t0=1$ .

Now, by (3.6), we have

$m ∞ ≤ lim k → + ∞ ⁡ I ∞ ⁢ ( t k ⁢ u k )$

$= lim k → + ∞ ⁡ ( I V ⁢ ( t k ⁢ u k ) + t k n n ⁢ ∫ ℝ n ( V ∞ - V ⁢ ( x ) ) ⁢ u k n ⁢ 𝑑 x )$

$= lim k → + ∞ ⁡ I V ⁢ ( t k ⁢ u k )$

$= lim k → + ∞ ⁡ t k n ⁢ ( ∫ ℝ n ( | ∇ ⁡ u k | n + V ⁢ ( x ) ⁢ u k n ) ⁢ 𝑑 x - ∫ ℝ n F ⁢ ( t k ⁢ u k ) t k n ⁢ u k n ⁢ u k n ⁢ 𝑑 x ) .$

This together with the monotonicity of $F⁢(t)tn$ (by the (A-R) condition) and $limk→∞⁡tk=1$ gives

$m ∞ ≤ lim k → + ∞ ⁡ I V ⁢ ( u k ) = m V ,$

which contradicts (3.3). This accomplishes the proof of Lemma 3.2. ∎

In the next result, we will exclude the possibilities of vanishing and dichotomy cases of the minimizing sequence in the Nehari manifold. For this purpose, we use a non-symmetric argument which is much easier than the one used in [8]. This argument has also been used in the subelliptic setting [10] and we modify that to apply to the Euclidean space here.

Lemma 3.3.

There holds

$lim k → + ∞ ⁡ ∫ ℝ n f ⁢ ( u k ) ⁢ u k ⁢ 𝑑 x = lim L → + ∞ ⁡ lim k → + ∞ ⁡ ∫ B L ⁢ ( 0 ) f ⁢ ( u k ) ⁢ u k ⁢ 𝑑 x .$

Proof.

Since $uk∈𝒩V$ , we have

$∫ ℝ n ( | ∇ ⁡ u k | n + V ⁢ ( x ) ⁢ u k n ) ⁢ 𝑑 x = lim L → + ∞ ⁡ lim k → + ∞ ⁡ ∫ B L ⁢ ( 0 ) f ⁢ ( u k ) ⁢ u k ⁢ 𝑑 x + lim L → + ∞ ⁡ lim k → + ∞ ⁡ ∫ ℝ n ∖ B L ⁢ ( 0 ) f ⁢ ( u k ) ⁢ u k ⁢ 𝑑 x$

$= : M 0 + M ∞ .$

We first show that $(M0,M∞)=(M,0)$ or $(M0,M∞)=(0,M)$ , where

$M = lim k → + ∞ ⁡ ∫ ℝ n f ⁢ ( u k ) ⁢ u k ⁢ 𝑑 x .$

Notice that we can write $∫ℝn(|∇⁡uk|n+V⁢(x)⁢ukn)⁢𝑑x$ as

$∫ ℝ n ( | ∇ ⁡ u k | n + V ⁢ ( x ) ⁢ u k n ) ⁢ 𝑑 x$

$= lim L → + ∞ ⁡ lim k → + ∞ ⁡ ∫ B L ⁢ ( 0 ) ( | ∇ ⁡ u k | n + V ⁢ ( x ) ⁢ u k n ) ⁢ 𝑑 x + lim L → + ∞ ⁡ lim k → + ∞ ⁡ ∫ ℝ n ∖ B L ⁢ ( 0 ) ( | ∇ ⁡ u k | n + V ⁢ ( x ) ⁢ u k n ) ⁢ 𝑑 x .$

Then we can assume that

$lim L → + ∞ ⁡ lim k → + ∞ ⁡ ∫ B L ⁢ ( 0 ) ( | ∇ ⁡ u k | n + V ⁢ ( x ) ⁢ u k n ) ⁢ 𝑑 x ≤ lim L → + ∞ ⁡ lim k → + ∞ ⁡ ∫ B L ⁢ ( 0 ) f ⁢ ( u k ) ⁢ u k ⁢ 𝑑 x$

(3.12)

$lim L → + ∞ ⁡ lim k → + ∞ ⁡ ∫ ℝ n ∖ B L ⁢ ( 0 ) ( | ∇ ⁡ u k | n + V ⁢ ( x ) ⁢ u k n ) ⁢ 𝑑 x ≤ lim L → + ∞ ⁡ lim k → + ∞ ⁡ ∫ ℝ n ∖ B L ⁢ ( 0 ) f ⁢ ( u k ) ⁢ u k ⁢ 𝑑 x .$

For a fixed $L>0$ , we define the function

$ϕ L 0 ⁢ ( x ) = { 1 if ⁢ | x | ≤ L , 0 ∼ 1 if ⁢ L < | x | ≤ L + 1 , 0 if ⁢ | x | ≥ L + 1 ,$

and $ϕL∞⁢(x)=1-ϕL0⁢(x)$ . Defining $uk,L*=uk⁢ϕL*$ ( $*=0$ or $∞$ ), one can easily derive that

$lim L → + ∞ ⁡ lim k → + ∞ ⁡ ∫ ℝ n ( | ∇ ⁡ u k , L 0 | n + V ⁢ ( x ) ⁢ ( u k , L 0 ) n ) ⁢ 𝑑 x = lim L → + ∞ ⁡ lim k → + ∞ ⁡ ∫ B L ⁢ ( 0 ) ( | ∇ ⁡ u k | n + V ⁢ ( x ) ⁢ u k n ) ⁢ 𝑑 x ,$

$lim L → + ∞ ⁡ lim k → + ∞ ⁡ ∫ ℝ n ( | ∇ ⁡ u k , L ∞ | n + V ⁢ ( x ) ⁢ ( u k , L ∞ ) n ) ⁢ 𝑑 x = lim L → + ∞ ⁡ lim k → + ∞ ⁡ ∫ ℝ n ∖ B L ⁢ ( 0 ) ( | ∇ ⁡ u k | n + V ⁢ ( x ) ⁢ u k n ) ⁢ 𝑑 x ,$

$lim L → + ∞ ⁡ lim k → + ∞ ⁡ ∫ ℝ n f ⁢ ( u k , L 0 ) ⁢ u k , L 0 ⁢ 𝑑 x = lim L → + ∞ ⁡ lim k → + ∞ ⁡ ∫ B L ⁢ ( 0 ) f ⁢ ( u k ) ⁢ u k ⁢ 𝑑 x ,$

$lim L → + ∞ ⁡ lim k → + ∞ ⁡ ∫ ℝ n f ⁢ ( u k , L ∞ ) ⁢ u k , L ∞ ⁢ 𝑑 x = lim L → + ∞ ⁡ lim k → + ∞ ⁡ ∫ ℝ n ∖ B L ⁢ ( 0 ) f ⁢ ( u k ) ⁢ u k ⁢ 𝑑 x ,$

$lim L → + ∞ ⁡ lim k → + ∞ ⁡ ∫ ℝ n F ⁢ ( u k , L 0 ) ⁢ 𝑑 x = lim L → + ∞ ⁡ lim k → + ∞ ⁡ ∫ B L ⁢ ( 0 ) F ⁢ ( u k ) ⁢ 𝑑 x ,$

$lim L → + ∞ ⁡ lim k → + ∞ ⁡ ∫ ℝ n F ⁢ ( u k , L ∞ ) ⁢ 𝑑 x = lim L → + ∞ ⁡ lim k → + ∞ ⁡ ∫ ℝ n ∖ B L ⁢ ( 0 ) F ⁢ ( u k ) ⁢ 𝑑 x .$

Without loss of generality, we assume that

(3.13)

$lim L → + ∞ ⁡ lim k → + ∞ ⁡ ∫ ℝ n ( | ∇ ⁡ u k , L 0 | n + V ⁢ ( x ) ⁢ ( u k , L 0 ) n ) ⁢ 𝑑 x ≤ lim L → + ∞ ⁡ lim k → + ∞ ⁡ ∫ ℝ n f ⁢ ( u k , L 0 ) ⁢ u k , L 0 ⁢ 𝑑 x .$

Then there exists some $tk,L0$ such that $tk,L0⁢uk,L0∈𝒩V$ , and satisfying

(3.14)

$lim L → + ∞ ⁡ lim k → + ∞ ⁡ t k , L 0 ≤ 1 .$

If $tk,L0≤1$ , then

$I V ⁢ ( t k , L 0 ⁢ u k , L 0 ) = 1 n ⁢ ∫ ℝ n ( f ⁢ ( t k , L 0 ⁢ u k , L 0 ) ⁢ t k , L 0 ⁢ u k , L 0 - n ⁢ F ⁢ ( t k , L 0 ⁢ u k , L 0 ) ) ⁢ 𝑑 x$

(3.15)

$≤ 1 n ⁢ ∫ ℝ n ( f ⁢ ( u k , L 0 ) ⁢ u k , L 0 - n ⁢ F ⁢ ( u k , L 0 ) ) ⁢ 𝑑 x .$

If $tk,L0≥1$ , then

$I V ⁢ ( t k , L 0 ⁢ u k , L 0 ) = 1 n ⁢ ( t k , L 0 ) n ⁢ ∫ ℝ n ( | ∇ ⁡ u k , L 0 | n + V ⁢ ( x ) ⁢ ( u k , L 0 ) n ) ⁢ 𝑑 x - ∫ ℝ n F ⁢ ( t k , L 0 ⁢ u k , L 0 ) ⁢ 𝑑 x$

(3.16)

$≤ 1 n ⁢ ( t k , L 0 ) n ⁢ ∫ ℝ n ( | ∇ ⁡ u k , L 0 | n + V ⁢ ( x ) ⁢ ( u k , L 0 ) n ) ⁢ 𝑑 x - ∫ ℝ n F ⁢ ( u k , L 0 ) ⁢ 𝑑 x .$

Combining (3.14)–(3.16) with (3.13), one has

$m V ≤ lim L → + ∞ ⁡ lim k → + ∞ ⁡ I V ⁢ ( t k , L 0 ⁢ u k , L 0 )$

$≤ 1 n ⁢ lim L → + ∞ ⁡ lim k → + ∞ ⁡ ∫ ℝ n ( f ⁢ ( u k , L 0 ) ⁢ u k , L 0 - n ⁢ F ⁢ ( u k , L 0 ) ) ⁢ 𝑑 x$

$≤ lim k → + ∞ ⁡ I V ⁢ ( u k ) = m V .$

Thus, we immediately have

$lim L → + ∞ ⁡ lim k → + ∞ ⁡ ∫ B L ⁢ ( 0 ) ( f ⁢ ( u k ) ⁢ u k - n ⁢ F ⁢ ( u k ) ) ⁢ 𝑑 x = m V$

and

$lim L → + ∞ ⁡ lim k → + ∞ ⁡ ∫ ℝ n ∖ B L ⁢ ( 0 ) ( f ⁢ ( u k ) ⁢ u k - n ⁢ F ⁢ ( u k ) ) ⁢ 𝑑 x = 0 .$

From the above and from the (A-R) condition, we can obtain

$M ∞ = lim L → + ∞ ⁡ lim k → + ∞ ⁡ ∫ ℝ n ∖ B L ⁢ ( 0 ) f ⁢ ( u k ) ⁢ u k ⁢ 𝑑 x = 0 ,$

and therefore $(M0,M∞)=(M,0)$ . Similarly, we can obtain $(M0,M∞)=(0,M)$ under assumption (3.12). From Lemma 3.2, we know $u≠0$ . Then $(M0,M∞)=(0,M)$ will not occur, and the proof of Lemma 3.3 is finished. ∎

Lemma 3.4.

There holds

$lim k → + ∞ ⁡ ∫ ℝ n F ⁢ ( u k ) ⁢ 𝑑 x = ∫ ℝ n F ⁢ ( u ) ⁢ 𝑑 x .$

Proof.

It follows from Lemma 3.3 that

$lim L → + ∞ ⁡ lim k → + ∞ ⁡ ∫ ℝ n ∖ B L ⁢ ( 0 ) f ⁢ ( u k ) ⁢ u k ⁢ 𝑑 x = 0 ,$

which together with the (A-R) condition implies that

$lim L → + ∞ ⁡ lim k → + ∞ ⁡ ∫ ℝ n ∖ B L ⁢ ( 0 ) F ⁢ ( u k ) ⁢ 𝑑 x = 0 .$

In order to obtain the desired convergence, we only need to prove that

$lim L → + ∞ ⁡ lim k → + ∞ ⁡ ∫ B L ⁢ ( 0 ) F ⁢ ( u k ) ⁢ 𝑑 x = ∫ ℝ n F ⁢ ( u ) ⁢ 𝑑 x .$

Indeed, for any $s>0$ , we have

$| ∫ B L ⁢ ( 0 ) F ⁢ ( u k ) ⁢ 𝑑 x - ∫ B L ⁢ ( 0 ) F ⁢ ( u ) ⁢ 𝑑 x |$

$≤ | ∫ B L ( 0 ) ∩ { | u k | < s } F ⁢ ( u k ) ⁢ 𝑑 x - ∫ B L ( 0 ) ∩ { | u k | < s } F ⁢ ( u ) ⁢ 𝑑 x | + | ∫ B L ( 0 ) ∩ { | u k | ≥ s } F ⁢ ( u k ) ⁢ 𝑑 x - ∫ B L ( 0 ) ∩ { | u k | ≥ s } F ⁢ ( u ) ⁢ 𝑑 x |$

$= I k , L , s + I ⁢ I k , L , s .$

A direct application of the dominated convergence theorem leads to $Ik,L,s→0$ . For $I⁢Ik,L,s$ , from condition (ii), we have $F⁢(s)≤c⁢f⁢(s)$ . Then it follows that

$∫ B L ( 0 ) ∩ { u k ≥ s } F ⁢ ( u k ) ⁢ 𝑑 x ≤ c s ⁢ ∫ ℝ n ∩ { u k ≥ s } f ⁢ ( u k ) ⁢ u k ⁢ 𝑑 x = c s ⁢ ∫ ℝ n f ⁢ ( u k ) ⁢ u k ⁢ 𝑑 x → 0 as ⁢ s → ∞ ,$

where we have used the fact that $∫ℝnf⁢(uk)⁢uk⁢𝑑x$ is bounded. Consequently, $I⁢Ik,L,s→0$ , and the proof of the lemma is finished. ∎

Lemma 3.5.

$∫ ℝ n ( | ∇ ⁡ u | n + V ⁢ ( x ) ⁢ | u | n ) ⁢ 𝑑 x > ∫ ℝ n f ⁢ ( u ) ⁢ u ⁢ 𝑑 x ,$

then

$lim k → + ∞ ⁡ ∫ ℝ n f ⁢ ( u k ) ⁢ u k ⁢ 𝑑 x = ∫ ℝ n f ⁢ ( u ) ⁢ u ⁢ 𝑑 x .$

Proof.

By Lemma 3.3, we only need to prove that

$lim L → + ∞ ⁡ lim k → + ∞ ⁡ ∫ B L ⁢ ( 0 ) f ⁢ ( u k ) ⁢ u k ⁢ 𝑑 x = ∫ ℝ n f ⁢ ( u ) ⁢ u ⁢ 𝑑 x .$

It follows from the lower semicontinuity of the norm in $W1,n⁢(ℝn)$ that

$lim k → ∞ ⁡ ∫ ℝ n ( | ∇ ⁡ u k | n + V ⁢ ( x ) ⁢ u k n ) ⁢ 𝑑 x ≥ ∫ ℝ n ( | ∇ ⁡ u | n + V ⁢ ( x ) ⁢ u n ) ⁢ 𝑑 x .$

We divide the proof into the following case.

Case 1. If

$∫ ℝ n ( | ∇ ⁡ u k | n + V ⁢ ( x ) ⁢ u k n ) ⁢ 𝑑 x = ∫ ℝ n ( | ∇ ⁡ u | n + V ⁢ ( x ) ⁢ u n ) ⁢ 𝑑 x ,$

then by the equivalence of norms we have $uk→u$ in $W1,n⁢(ℝn)$ , and $uk→u$ in $Lp⁢(ℝn)$ for any $p≥n$ . From the Trudinger–Moser inequality (1.3) in $W1,n⁢(ℝn)$ , we see that for any $p0>1$ ,

$sup k ⁡ ∫ ℝ n ( f ⁢ ( u k ) ⁢ u k ) p 0 ⁢ 𝑑 x < ∞ ,$

which implies that

$lim L → + ∞ ⁡ lim k → ∞ ⁡ ∫ B L f ⁢ ( u k ) ⁢ u k ⁢ 𝑑 x = ∫ ℝ n f ⁢ ( u ) ⁢ u ⁢ 𝑑 x .$

Case 2. If

$lim k → ∞ ⁡ ∫ ℝ n ( | ∇ ⁡ u k | n + V ⁢ ( x ) ⁢ u k n ) ⁢ 𝑑 x > ∫ ℝ n ( | ∇ ⁡ u | n + V ⁢ ( x ) ⁢ u n ) ⁢ 𝑑 x ,$

we set

$v k := u k lim k → ∞ ⁡ ∥ u k ∥ W V 1 , n ⁢ ( ℝ n ) and v 0 := u lim k → ∞ ⁡ ∥ u k ∥ W V 1 , n ⁢ ( ℝ n ) .$

We claim that there exists $q0>1$ such that

$q 0 ⁢ ∥ u k ∥ W V 1 , n ⁢ ( ℝ n ) n < ( α n / α 0 ) n - 1 1 - ∥ v 0 ∥ W V 1 , n ⁢ ( ℝ n ) n .$

Indeed, by the (A-R) condition (ii), we have

$I V ⁢ ( u ) = 1 n ⁢ ∫ ℝ n ( | ∇ ⁡ u | n + V ⁢ ( x ) ⁢ u n ) ⁢ 𝑑 x - ∫ ℝ n F ⁢ ( u ) ⁢ 𝑑 x$

$> 1 n ⁢ ∫ ℝ n ( f ⁢ ( u ) - n ⁢ F ⁢ ( u ) ) ⁢ 𝑑 x > 0 .$

Hence, from Lemma 3.1 and Lemma 3.4, we get

$lim k → ∞ ⁡ ( ∥ u k ∥ W V 1 , n ⁢ ( ℝ n ) n ) ⁢ ( 1 - ∥ v 0 ∥ W V 1 , n ⁢ ( ℝ n ) n ) = lim k → ∞ ⁡ ( ∥ u k ∥ W V 1 , n ⁢ ( ℝ n ) n ) ⁢ ( 1 - ∥ u ∥ W V 1 , n ⁢ ( ℝ n ) n ∥ u k ∥ W V 1 , n ⁢ ( ℝ n ) )$

$= n ⁢ m V + n ⁢ ∫ ℝ n F ⁢ ( u k ) ⁢ 𝑑 x - n ⁢ I V ⁢ ( u ) - n ⁢ ∫ ℝ n F ⁢ ( u ) ⁢ 𝑑 x$

$< ( α n α 0 ) n - 1 .$

Combining the above estimate with the concentration compactness principle for the Trudinger–Moser inequality (1.7), which can be established similarly to [35, Theorem 5.2], one can derive that there exists $p0>1$ such that

$sup k ⁡ ∫ ℝ n ( f ⁢ ( u k ) ⁢ u k ) p 0 ⁢ 𝑑 x < ∞ .$

Then the Vitali convergence theorem implies that

$lim L → + ∞ ⁡ lim k → ∞ ⁡ ∫ B L f ⁢ ( u k ) ⁢ u k ⁢ 𝑑 x = ∫ ℝ n f ⁢ ( u ) ⁢ u ⁢ 𝑑 x ,$

which finishes the proof of Lemma 3.5. ∎

Now, we give the proof of Theorem 1.3.

Existence of Ground-State Solutions:

We will prove that $mV$ is achieved by u. We claim that

(3.17)

$∥ u ∥ W V 1 , n ⁢ ( ℝ n ) n ≤ ∫ ℝ n f ⁢ ( u ) ⁢ u ⁢ 𝑑 x .$

We prove this claim by contradiction. If

$∥ u ∥ W V 1 , n ⁢ ( ℝ n ) n > ∫ ℝ n f ⁢ ( u ) ⁢ u ⁢ 𝑑 x ,$

by Lemma 3.5 and Lemma 3.1, we can deduce

$lim k → ∞ ⁡ ∫ ℝ n f ⁢ ( u k ) ⁢ u k ⁢ 𝑑 x = ∫ ℝ n f ⁢ ( u ) ⁢ u ⁢ 𝑑 x .$

This implies that

$∥ u ∥ W V 1 , n ⁢ ( ℝ n ) n ≤ lim k → ∞ ⁡ ∥ u k ∥ W V 1 , n ⁢ ( ℝ n ) n = lim k → ∞ ⁡ ∫ ℝ n f ⁢ ( u k ) ⁢ u k ⁢ 𝑑 x = ∫ ℝ n f ⁢ ( u ) ⁢ u ⁢ 𝑑 x < ∥ u ∥ W V 1 , n ⁢ ( ℝ n ) n ,$

which is a contradiction.

By (3.17), there exists $γ0∈(0,1]$ such that $γ0⁢u∈𝒩V$ . Then we can derive that

$m V ≤ I ⁢ ( γ 0 ⁢ u )$

$= 1 n ⁢ ∫ ℝ n ( f ⁢ ( γ 0 ⁢ u ) ⁢ ( γ 0 ⁢ u ) - n ⁢ F ⁢ ( γ 0 ⁢ u ) ) ⁢ 𝑑 x$

$≤ 1 n ⁢ ∫ ℝ n ( f ⁢ ( u ) ⁢ ( u ) - n ⁢ F ⁢ ( u ) ) ⁢ 𝑑 x$

$≤ lim k → ∞ ⁡ 1 n ⁢ ∫ ℝ n ( f ⁢ ( u k ) ⁢ ( u k ) - n ⁢ F ⁢ ( u k ) ) ⁢ 𝑑 x$

$= lim k → ∞ ⁡ I V ⁢ ( u k ) = m V ,$

which means that $γ0=1$ , $u∈𝒩V$ and $IV⁢(u)=mV$ , and the proof of Theorem 1.3 is finished.

Communicated by Laurent Veron

Funding source: National Natural Science Foundation of China

Award Identifier / Grant number: 11901031

Award Identifier / Grant number: 12071185

Funding source: Simons Foundation

Award Identifier / Grant number: 519099

Funding statement: The first author was partly supported by the National Natural Science Foundation of China (No. 11901031). The second author was partly supported by a Simons grant from the Simons Foundation. The third author was partly supported by the National Natural Science Foundation of China (No. 12071185).

Acknowledgements

The authors wish to thank the referee for his/her very careful reading of the paper and useful comments which have improved the exposition of this paper.

References

[1] S. Adachi and K. Tanaka, Trudinger type inequalities in $𝐑N$ and their best exponents, Proc. Amer. Math. Soc. 128 (2000), no. 7, 2051–2057. 10.1090/S0002-9939-99-05180-1Search in Google Scholar

[2] C. O. Alves and G. M. Figueiredo, On multiplicity and concentration of positive solutions for a class of quasilinear problems with critical exponential growth in $ℝN$ , J. Differential Equations 246 (2009), no. 3, 1288–1311. 10.1016/j.jde.2008.08.004Search in Google Scholar

[3] C. O. Alves, M. A. S. Souto and M. Montenegro, Existence of a ground state solution for a nonlinear scalar field equation with critical growth, Calc. Var. Partial Differential Equations 43 (2012), no. 3–4, 537–554. 10.1007/s00526-011-0422-ySearch in Google Scholar

[4] A. Ambrosetti and A. Malchiodi, Perturbation Methods and Semilinear Elliptic Problems on $𝐑n$ , Progr. Math. 240, Birkhäuser, Basel, 2006. 10.1007/3-7643-7396-2Search in Google Scholar

[5] D. M. Cao, Nontrivial solution of semilinear elliptic equation with critical exponent in $𝐑2$ , Comm. Partial Differential Equations 17 (1992), no. 3–4, 407–435. 10.1080/03605309208820848Search in Google Scholar

[6] L. Chen, J. Li, G. Lu and C. Zhang, Sharpened Adams inequality and ground state solutions to the bi-Laplacian equation in $ℝ4$ , Adv. Nonlinear Stud. 18 (2018), no. 3, 429–452. 10.1515/ans-2018-2020Search in Google Scholar

[7] L. Chen, G. Lu and M. Zhu, Ground states of bi-harmonic equations with critical exponential growth involving constant and trapping potentials, Calc. Var. Partial Differential Equations 59 (2020), no. 6, Paper No. 185. 10.1007/s00526-020-01831-4Search in Google Scholar

[8] L. Chen, G. Lu and M. Zhu, A critical Trudinger–Moser inequality involving a degenerate potential and nonlinear Schrödinger equations, Sci. China Math. 64 (2021), no. 7, 1391–1410. 10.1007/s11425-020-1872-xSearch in Google Scholar

[9] L. Chen, G. Lu and M. Zhu, Existence and non-existence of ground states of bi-harmonic equations involving constant and degenerate Rabinowitz potentials, preprint (2021), https://arxiv.org/abs/2108.06301. 10.1007/s00526-022-02375-5Search in Google Scholar

[10] L. Chen, G. Lu and M. Zhu, Least energy solutions to quasilinear subelliptic equations with constant and degenerate potentials on the Heisenberg group, preprint. 10.1112/plms.12495Search in Google Scholar

[11] L. Chen, G. Lu and M. Zhu, Sharp Trudinger–Moser inequality with a degenerate potential on complete and noncompact Riemannian manifolds and quasilinear Schrödinger equations, preprint. Search in Google Scholar

[12] W. Y. Ding and W.-M. Ni, On the existence of positive entire solutions of a semilinear elliptic equation, Arch. Ration. Mech. Anal. 91 (1986), no. 4, 283–308. 10.1007/BF00282336Search in Google Scholar

[13] J. M. do Ó, M. de Souza and E. de Medeiros, An improvement for the Trudinger–Moser inequality and applications, J. Differential Equations 256 (2014), 1317–1349. 10.1016/j.jde.2013.10.016Search in Google Scholar

[14] S. Ibrahim, N. Masmoudi and K. Nakanishi, Trudinger–Moser inequality on the whole plane with the exact growth condition, J. Eur. Math. Soc. (JEMS) 17 (2015), no. 4, 819–835. 10.4171/JEMS/519Search in Google Scholar

[15] N. Lam and G. Lu, Existence and multiplicity of solutions to equations of N-Laplacian type with critical exponential growth in $ℝN$ , J. Funct. Anal. 262 (2012), no. 3, 1132–1165. 10.1016/j.jfa.2011.10.012Search in Google Scholar

[16] N. Lam and G. Lu, Sharp Moser–Trudinger inequality on the Heisenberg group at the critical case and applications, Adv. Math. 231 (2012), no. 6, 3259–3287. 10.1016/j.aim.2012.09.004Search in Google Scholar

[17] N. Lam and G. Lu, A new approach to sharp Moser–Trudinger and Adams type inequalities: A rearrangement-free argument, J. Differential Equations 255 (2013), no. 3, 298–325. 10.1016/j.jde.2013.04.005Search in Google Scholar

[18] J. Li and G. Lu, Critical and subcritical Trudinger–Moser inequalities on complete noncompact Riemannian manifolds, Adv. Math. 389 (2021), Paper No. 107915. 10.1016/j.aim.2021.107915Search in Google Scholar

[19] J. Li, G. Lu and M. Zhu, Concentration-compactness principle for Trudinger–Moser inequalities on Heisenberg groups and existence of ground state solutions, Calc. Var. Partial Differential Equations 57 (2018), no. 3, Paper No. 84. 10.1007/s00526-018-1352-8Search in Google Scholar

[20] Y. Li, Moser–Trudinger inequality on compact Riemannian manifolds of dimension two, J. Partial Differential Equations 14 (2001), no. 2, 163–192. Search in Google Scholar

[21] Y. Li and B. Ruf, A sharp Trudinger–Moser type inequality for unbounded domains in $ℝn$ , Indiana Univ. Math. J. 57 (2008), no. 1, 451–480. 10.1512/iumj.2008.57.3137Search in Google Scholar

[22] P.-L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. I, Ann. Inst. H. Poincaré Anal. Non Linéaire 1 (1984), no. 2, 109–145. 10.1016/s0294-1449(16)30428-0Search in Google Scholar

[23] G. Lu and H. Tang, Best constants for Moser–Trudinger inequalities on high dimensional hyperbolic spaces, Adv. Nonlinear Stud. 13 (2013), no. 4, 1035–1052. 10.1515/ans-2013-0415Search in Google Scholar

[24] G. Lu and H. Tang, Sharp singular Trudinger–Moser inequalities in Lorentz–Sobolev spaces, Adv. Nonlinear Stud. 16 (2016), no. 3, 581–601. 10.1515/ans-2015-5046Search in Google Scholar

[25] G. Lu and J. Wei, On nonlinear Schrödinger equations with totally degenerate potentials, C. R. Acad. Sci. Paris Sér. I Math. 326 (1998), no. 6, 691–696. 10.1016/S0764-4442(98)80032-3Search in Google Scholar

[26] N. Masmoudi and F. Sani, Trudinger–Moser inequalities with the exact growth condition in $ℝN$ and applications, Comm. Partial Differential Equations 40 (2015), no. 8, 1408–1440. 10.1080/03605302.2015.1026775Search in Google Scholar

[27] J. Moser, A sharp form of an inequality by N. Trudinger, Indiana Univ. Math. J. 20 (1970/71), 1077–1092. 10.1512/iumj.1971.20.20101Search in Google Scholar

[28] S. I. Pohozhaev, The Sobolev embedding in the case $p⁢l=n$ , Proceedings of the Technical Science Conference on Advanced Science Research 1964-1965, Mathematics Section, Moscow Nerger Institute, Moscow (1965), 158–170. Search in Google Scholar

[29] P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Reg. Conf. Ser. Math. 65, American Mathematical Society, Providence, 1986. 10.1090/cbms/065Search in Google Scholar

[30] P. H. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys. 43 (1992), no. 2, 270–291. 10.1007/BF00946631Search in Google Scholar

[31] B. Ruf and F. Sani, Ground states for elliptic equations in $ℝ2$ with exponential critical growth, Geometric Properties for Parabolic and Elliptic PDE’s, Springer INdAM Ser. 2, Springer, Milan (2013), 251–267. 10.1007/978-88-470-2841-8_16Search in Google Scholar

[32] N. S. Trudinger, On imbeddings into Orlicz spaces and some applications, J. Math. Mech. 17 (1967), 473–483. 10.1512/iumj.1968.17.17028Search in Google Scholar

[33] X. Wang, On concentration of positive bound states of nonlinear Schrödinger equations, Comm. Math. Phys. 153 (1993), no. 2, 229–244. 10.1007/BF02096642Search in Google Scholar

[34] C. Zhang and L. Chen, Concentration-compactness principle of singular Trudinger–Moser inequalities in $ℝn$ and n-Laplace equations, Adv. Nonlinear Stud. 18 (2018), no. 3, 567–585. 10.1515/ans-2017-6041Search in Google Scholar

[35] M. C. Zhu, J. Wang and X. Y. Qian, Existence of solutions to nonlinear Schrödinger equations involving N-Laplacian and potentials vanishing at infinity, Acta Math. Sin. (Engl. Ser.) 36 (2020), no. 10, 1151–1170. 10.1007/s10114-020-0020-zSearch in Google Scholar

Received: 2021-07-01

Revised: 2021-09-12

Accepted: 2021-09-14

Published Online: 2021-10-15

Published in Print: 2021-11-01

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

Articles in the same Issue

DOI: doi.org/10.1515/ans-2021-2146

Keywords for this article

Sharp Trudinger–Moser Inequalities; Degenerate Potential; Ground State Solutions; Schrödinger Equations; Nehari Manifold

Creative Commons

BY 4.0