Processing math: 100%
Home Ground states for Schrödinger-Poisson system with zero mass and the Coulomb critical exponent
Article Open Access

Ground states for Schrödinger-Poisson system with zero mass and the Coulomb critical exponent

  • Jing Zhang , Dongdong Qin , Siti Sahara and Qingfang Wu EMAIL logo
Published/Copyright: March 13, 2025

Abstract

This article focuses on the study of the following Schrödinger-Poisson system with zero mass:

{Δu+ϕu=uu+f(u),xR3,Δϕ=u2,xR3,

where f is a continuous function satisfying some general growth conditions, and it requires only to be super-quadratic growth at infinity and includes, in particular, the pure power function up2u with p(3,6) . The nonlinear term uu is the so-called Coulomb critical nonlinearity because it presents a certain scaling invariance and the mountain-pass geometry cannot be established when f=0 . Few results are known to such case. By developing some delicate analyses and using detailed estimates, we obtain the existence of ground-states and least energy solution for the aforementioned system under some natural assumptions on f .

MSC 2010: 35J20; 35J60

1 Introduction and main results

This article is dedicated to studying the existence of ground-state solution for the following Schrödinger-Poisson system with zero mass and the Coulomb critical exponent:

(1.1) {Δu+ϕu=uu+f(u),xR3,Δϕ=u2,xR3,

where the nonlinearity f satisfies the following basic assumptions:

  1. fC(R,R) and

    limt0f(t)t2=0,limt+f(t)t5=0;

  2. there holds

    limt+F(t)t3=+,

    where F(t)t0f(s)ds ;

  3. for every τ0 , f(τ)τ0 , and the function t2f(tτ)(tτ)3F(tτ)t3 is increasing on t(0,+) ;

  4. f(t)t>3F(t)>0,tR\{0} , and there exist R0>0,κ(32,3] , and C0>0 such that

    f(t)tR0f(t)tκC0[f(t)t3F(t)].

Define

(1.2) ϕu(x)14πxu2=R3u2(y)4πxydy,xR3.

As we know, Δϕu=u2 in a weak sense, and problem (1.1) can be deduced from the following equation:

(1.3) Δu+ωu+(u214πx)u=f(u),xR3.

This is the well-known Schrödinger-Poisson-Slater problem, and it stems from the Slater approximation of the exchange term in the Hartree-Fock model, which appears in quantum mechanics in the study of a system of N particles [35]. ω0 represents the phase of the standing wave for the time-dependent equation, the convolution term denotes the Coulombic repulsion between the electrons, and the function u2 presents the density of electrons. In such framework, f(u)=u23u was used by Slater to approximate the exchange term, and other exponents have been used in different approximations (see [3,27,30,32] for more information on these models). Such problems present a combination of repulsive forces (induced by the nonlocal term) and attractive forces (induced by the local term); the interaction between them leads to unexpected situations and also brings some interesting challenges in mathematics.

When ω>0 , the working space is the usual Sobolev space H1(R3) and fruitful results are obtained by researchers under various assumptions imposed on the nonlinearity f (see, e.g., [1,2,5,6,913,15,28,3234, 3638,42,43] and references therein); particularly, the existence or nonexistence of solutions, radial or nonradial solutions, positive or sign-changing solutions, bound-states and ground-states, semi-classical states, and normalized solutions are established there.

As pointed out by Ruiz [32], the behavior of radial minimizers motivates the study of the static case, i.e., ω=0 . Such case is called “zero mass” problem by Berestycki and Lions [4], because the linearized operator at zero involves only the Laplacian operator. Note that the absence of a phase term makes the space H1(R3) not to be a good framework for studying (1.3). The following space was first introduced by Ruiz [32]:

(1.4) E{uD1,2(R3):R3R3u2(x)u2(y)xydxdy<},

and it was shown by Ruiz [32, Proposition 2.2] that such space is a uniformly convex Banach space with the following norm:

uE[R3u2dx+(R3R3u2(x)u2(y)4πxydxdy)12]12,uE.

The double integral expression is the so-called Coulomb energy of wave and has been well studied by Lieb and Loss in their classic book [26]. From this point of view, E is the subspace of D1,2(R3) such that the Coulomb energy of the charge is finite. Denote by Er the subspace of radial functions. Ruiz [32] obtained a general lower bound for the Coulomb energy, taking advantage of which useful embedding result was established there, i.e., ErLs(R3) is continuous for p(187,6] , and the inclusion is compact for p(187,6) . This explains the significance of the exponent 187 in the radial case. Moreover, the existence of positive solution was shown there [32] for the following equation with p(187,3) :

(1.5) Δu+(u214πx)u=μup2u,xR3,μ>0.

Later, Ianni and Ruiz [16] considered the case p(3,6) and obtained the existence of ground-states and infinitely many radial bound-state solutions using the monotonicity trick and Krasnoselskii genus (see also [20] for similar results). Using the Pohozaev identity, we know that (1.5) has no solution in EH2loc(R3) for p6 (cf. [16]). Liu et al. [29] first considered the following problem with Sobolev critical exponent and lower-order perturbation:

(1.6) Δu+(u214πx)u=μup2u+u5,xR3,

where μ>0 and 187<p<6 . Moreover, they showed that (1.1) has a positive solution for p(4,6) and p(3,4] with μ large enough via a perturbation procedure. It is worth pointing out that the effect of the additive perturbation is to lower the energy such that the ground-state level of the functional can be controlled by a fine threshold. For similar results, we refer readers to [21,22]. These results were improved recently in previous studies [14,24] where some existence results were obtained by developing a much simpler method. For critical Schrödinger-Poisson-Slater problem (1.6) with non-autonomous nonlinearity, we refer readers to [41,44,45] where existence and multiplicity of positive solutions were established there. For other related results of nonlocal problems, we draw reader’s attention to previous studies [7,23,39].

The case p=3 is called Coulomb critical since it exhibits a certain invariance, i.e., for a solution u of (1.5) and parameter λR , the function λ2u(λx) is also a solution of (1.5). It was shown by Ianni and Ruiz [16, Proposition 5.1] that (1.5) has only trivial solution for μ<2 and p=3 . Moreover, they obtained the following result.

Theorem 1.1

[16, Theorem 1.3] There exists an increasing sequence μk>0,μk+ such that the problem

Δu+(u214πx)u=μkuu

has a radial solution ukEr , where μk is the Lagrange multiplier which is not priori.

By Theorem 1.1, it is reasonable that the solutions in such case (i.e., p=3 in (1.5)) appear only for certain values of μ , and all possible solutions of (1.5) have energy equal to zero using Pohozaev-type equality (cf. [16, Remark 5.2]). We emphasize that we cannot get rid of the Lagrange multiplier μk ; this is not a problem of the method of the proof, but it is something intrinsic of the problem. In such case, the mountain-pass geometry cannot be verified by the energy functional corresponding to (1.5), and (1.5) should be interpreted as an eigenvalue problem rather than an equation (cf. [16,31]). In order to obtain more information about solutions for Schrödinger-Poisson-Slater problem with Coulomb critical exponent, it is reasonable to consider the nonlinearity as the combination of critical term uu and perturbation term f(u) , i.e., problem (1.1). Lei et al. [19] considered (1.1) with f(u)=μup2u , p(3,6) and showed that the Nehari-Pohozaev manifold is a C1 manifold, which is a natural constraint set for the corresponding energy functional. Existence of ground-state solution for (1.1) was obtained there by establishing a splitting lemma for a bounded Palais-Smale sequence on the Nehari-Pohozaev manifold. We point out that it is very crucial that the nonlinearity f considered there [19] is a pure power function, which guarantees the Nehari-Pohozaev set is a C1 manifold, and the Nehari-Pohozaev method used there becomes invalid without the C1 smooth restriction on the nonlinearity. For the following nonlinearity f , even though it possesses smoothness, we do not know whether problem (1.1) still has a nontrivial solution:

f(t)=11+ttp1t+pln(1+t)tp2t,3p<5.

It is nature to ask whether the result obtained in [19] still holds for general nonlinearity; precisely, the following question will be addressed in this article.

Question 1. Study problem (1.1) with Coulomb critical exponent and zero mass. In this case, how to obtain the ground-state solution and least energy solution without C1 smooth restriction on the nonlinearity.

Since we only require that the nonlinearity f is super-quadratic growth at infinity (i.e., (F2) holds), the main difficulty to obtain a ground-state solution of (1.1) lies in verifying the boundedness of a proper Palais-Smale sequence or Cerami sequence. Different from Theorem 1.1 (cf. [16, Theorem 1.3]) and [29, Theorem 1.4], we work in the space E instead of the radial space Er , and the compact embedding EL3(R3) does not hold in such case; thus, solutions obtained here is not limited to be radial. Our purpose in this article is twofold. First, we try to establish a ground-state solution of (1.1) under general assumptions (F1)–(F3); to this end, we employ the strategy used in [37] and apply the deformation lemma together with some delicate estimates. Second, we pay our attention on the existence of least energy solution for (1.1). We introduce a new condition (F4), which is much weaker than the Ambrosetti-Rabinowitz type condition and use a perturbation method employed by Jeanjean [17,18] to achieve our goal. Thus, Question 1 is well solved (see Examples 1 and 2 for functions satisfying (F1)–(F4)). Such existence results, to the best of our knowledge, are up to date, which partially extend and, in fact, complement the aforementioned results of previous studies [14,16,19,29,31,32].

As well known, the energy functional corresponding to (1.1) is

(1.7) Φ(u)=12R3u2dx+14R3R3u2(x)u2(y)4πxydxdyR3[13u3+F(u)]dx.

Define the “Nehari-Pohozaev” manifold introduced by Ruiz [33] by

(1.8) {uE\{0}:J(u)=0},

where

(1.9) J(u)=32u22+34R3R3u2(x)u2(y)4πxydxdyu33R3[2f(u)u3F(u)]dx.

Our main results in this article read as follows:

Theorem 1.2

Assume that (F1)–(F3) hold. Then, problem (1.1) has a ground-state solution of Nehari-Pohozaev type, i.e., a nontrivial solution ˉuE such that Φ(ˉu)=infΦ>0 .

Theorem 1.3

Assume that (F1), (F2), and (F4) hold. Then, problem (1.1) has a least energy solution u0E\{0} .

Before ending this section, we give two nonlinear examples: both of them satisfy assumptions (F1)–(F4).

Example 1.4

f(t)=μptp2t,andF(t)=μtp,

where 3<t<6 and μ>0 .

Example 1.5

f(t)=μ11+ttp1t+pμln(1+t)tp2t,andF(t)=μln(1+t)tp,

where 3p<5 and μ>0 .

Throughout this article, set ut(x)u(tx) for t>0 , and denote the norm of Ls(R3) by us=(R3usdx)1s for s1 . Br(x)={yR3:yx<r} and C1,C2, stand for positive constants, which may be different in different places.

2 Variational framework and preliminaries

In this section, we collect some basic notations and preliminaries, which will be used in the proofs of our main results.

Consider the space E defined by (1.4), and for any uE , set

(2.1) N[u]R3R3u2(x)u2(y)4πxydxdy,

(2.2) Q[u]2u22+N[u],

and

(2.3) uE[u22+N[u]]12.

In the following, we list some useful properties about the space E with the norm E (see, e.g., [16,29,32,38]).

Lemma 2.1

[32]   E is a norm, and (E,E) is a uniformly convex Banach space. Moreover, C0(R3) is dense in E, and ELs(R3) is continuous for p[3,6] .

Lemma 2.2

[38] Assume that a,b>0 . Then, there holds

(2.4) au22+bN[u]2abu33,uE.

For function ϕu defined by (1.2), we have ϕu(x)>0 when u0 , and uE if and only if both u,ϕuD1,2(R3) . Moreover, by Δϕ=u2 , we have

(2.5) R3ϕuvdx=R3u2vdx,vE,

and

(2.6) R3R3u2(x)u2(y)4πxydxdy=R3ϕu(x)u2dx.

Using Hardy-Littlewood-Sobolev inequality (see [25] or [26, page 98]), one has

(2.7) R3R3u(x)v(y)xydxdy83233πu65v65,u,vL65(R3).

Lemma 2.3

For any s[3,6] , there exists Cs>0 such that

(2.8) ussCs(Q[u])(2s3)3,uE.

Proof

This lemma has been proved in [16, Lemma 3.1]; here, we give a direct proof with some information of the embedding constant Cs . Observe that

2R3u3dx=R3ϕuudxR3ϕu2dx+R3u2dxQ[u].

In view of the Sobolev inequality [40], there exists a best embedding constant S>0 such that

(2.9) Su26u22,uD1,2(R3),

from which we deduce that

R3u6dx(2S)3(Q[u])3.

It follows from the Hölder inequality that

ussu6s3u2s66232s3S3s(Q[u])(2s3)3,uE,s[3,6].

By setting Cs232s3S3s , we obtain (2.8).□

Lemma 2.4

[32] Suppose that {un}E . Then,

  1. unˉu in E if and only if unˉu and ϕunϕˉu in D1,2(R3) ;

  2. unˉu in E if and only if unˉu in D1,2(R3) and supN[un]<+ . In such case, ϕunϕˉu in D1,2(R3) .

As in [16,32], we define functionals T and D as follows:

(2.10) T:E4R,T(u,v,w,z)R3R3u(x)v(x)w(y)z(y)4πxydxdy

and

(2.11) D:E2R,D(u,v)R3R3u(x)v(y)4πxydxdy.

Lemma 2.5

[16] Suppose that {un},{vn},{wn}E , zE . If unˉu,vnˉv,wnˉw in E, then

T(un,vn,wn,z)T(ˉu,ˉv,ˉw,z).

We have the following vanishing lemma due to Lions [27, Lemma I.1] (see [16,29]).

Lemma 2.6

If unˉu in E, and

(2.12) limnsupyRNB1(y)un3dx=0,

then

(2.13) uns0,s(3,6).

It follows from (F1) and Lemma 2.1 that the functional Φ defined by (1.7) is of class C1 , and

(2.14) Φ(u),v=R3uvdx+R3ϕu(x)uvdxR3(uu+f(u))vdx,u,vE.

It is not difficult to verify that solutions of (1.1) are critical points of the functional (1.7).

3 Existence of ground-state solution

Lemma 3.1

Assume that (F1) and (F3) hold. Then,

  1. for every τ0 , there holds

    (3.1) h(t,τ)t3F(t2τ)+2(1t3)3f(τ)τ(2t3)F(τ)>0,t(0,1)(1,+);

  2. there holds

    (3.2) f(τ)τ>3F(τ)>0,τ0.

Proof

(i). For any τ0 , by (F3) and a simple calculation, we have

(3.3) ddth(t,τ)=t2[2f(t2τ)(t2τ)2F(t2τ)t62f(τ)τ2F(τ)1]{>0,t(1,+),<0,t(0,1).

This shows that h(t,τ)>h(1,τ)=0,t(0,1)(1,+) ; thus, (3.1) holds. By (F1) and (3.3), one has limt0+h(t,τ)>0 .

(ii). Letting t0+ in (3.1) and using the fact limt0+h(t,τ)>0 , we have

(3.4) f(τ)τ3F(τ)>0,τ0.

It follows that (ii) holds.□

Lemma 3.2

Assume that (F1) and (F3) hold. Then,

(3.5) Φ(u)=Φ(t2ut)+1t33J(u)+R3h(t,u)dx,uE,t0.

Proof

Note that

(3.6) Φ(t2ut)=t32R3u2dx+t34R3R3u2(x)u2(y)4πxydxdyR3[t33u3+1t3F(t2u)]dx.

By (1.7), (1.9), and (3.6), one has

Φ(u)Φ(t2ut)=1t32R3u2dx+1t34R3R3u2(x)u2(y)4πxydxdy+R3[t313u3+1t3F(t2u)F(u)]dx=1t33J(u)+R3[2(1t3)3f(u)u+1t3F(t2u)(2t3)F(u)]dx=1t33J(u)+R3h(t,u)dx.

This shows that (3.5) holds.□

From Lemmas 3.1 and 3.2, we have the following corollary immediately.

Corollary 3.3

Assume that (F1) and (F3) hold. Then, for u ,

(3.7) Φ(u)=maxt0Φ(t2ut).

Lemma 3.4

Assume that (F1), (F2), and (F3) hold. Then, for any uE\{0} , there exists a tu>0 such that t2uutu .

Proof

Let uE\{0} be fixed and define a function ζ(t)Φ(t2ut) on [0,) . By (1.9) and (3.6), we have for any t>0 ,

ζ(t)=03t32u22+3t34N[u]t3u33t3R3[2f(t2u)(t2u)3F(t2u)]dx=0J(t2ut)=0t2ut.

By (F1), (2.4), and (3.6), we have

(3.8) ξ(t)=t32u22+t34N[u]t33u33t3R3F(t2u)dxt32u22+t34N[u]2t35u33C1t9u66t3[14u22+9100N[u]]C1t9u66,t>0.

From (3.8) and (F2), it is easy to verify that ζ(0)=0 , ζ(t)>0 for t>0 small and ζ(t)<0 for t large. Therefore, maxt[0,)ζ(t) is achieved at a tu=t(u)>0 so that ζ(tu)=0 and t2uutu .□

Both Corollary 3.3 and Lemma 3.4 imply the following lemma.

Lemma 3.5

Assume that (F1), (F2), and (F3) hold. Then,

infuΦ(u)m0=infuE\{0}maxt0Φ(t2ut).

Lemma 3.6

Assume that (F1), (F2), and (F3) hold. Then,

  1. there exists ρ0>0 such that u22ρ0,u ;

  2. m0=infuΦ(u)>0 .

Proof

Since J(u)=0,u , by (F1), (1.9), (2.4), and (2.9), it follows that

34u22+32u3332R3u2dx+34R3ϕu(x)u2dx=u33+R3[2f(u)u3F(u)]dx32u33+C1u6632u33+C1S3u62,

where C1 is a positive constant. This implies

(3.9) u22ρ03S322C1,u.

From (F1), (1.7), (2.4), (2.9), (3.6), and (3.7), we have

(3.10) Φ(u)Φ(t2ut)=t32u22+t34N[u]t33u33t3R3F(t2u)dxt32u22+t34N[u]2t35u33C2t9u6617t350u22C2t9S3u62,u.

For any u , by (3.9), we can set t=(17S3150C2u42)16>0 . Then, it follows from (3.10) that

Φ(u)17t350u22C2t9S3u621717S323756C2,u.

This shows that m0=infuΦ(u)>0 .□

In view of [37, Lemma 2.8] and [14, Lemma 3.9], we have the following lemma.

Lemma 3.7

Assume that (F1), (F2), and (F3) hold. If unˉu in E, then

(3.11) Φ(un)=Φ(ˉu)+Φ(unˉu)+o(1),

(3.12) Φ(un),un=Φ(ˉu),ˉu+Φ(unˉu),unˉu+o(1),

and

(3.13) J(un)=J(ˉu)+J(unˉu)+o(1).

Lemma 3.8

Assume that (F1), (F2), and (F3) hold. Then, m0 is achieved.

Proof

We prove this lemma by using the strategy used in [37]. Let {un} be such that Φ(un)m0 . We first prove that supnNQ[un]<+ . Otherwise, we may assume that limnQ[un]=+ . Let tn3Q[un] and vn(x)t2nun(xtn) . Then, Q[vn]=1 . Choose R=232m0 . If δlimsupnsupyR3B1(y)vn3dx>0 , then in light of Lemma 2.6, vns0 for s(3,6) . It follows from (F1), (2.4), (2.8), (3.6), and (3.7) that

(3.14) m0+o(1)=Φ(un)Φ((Rtn)2(un)Rtn)=Φ(R2(vn)R)=R32vn22+R34N[vn]R33vn331R3RF(R2vn)dxR38Q[vn]1R3RF(R2vn)dx=R38o(1)=2m0o(1).

This contradiction shows that δ>0 . Without loss of generality, we may assume the existence of ynR3 such that B1(yn)vn3dx>δ2 . Let ˆvn(x)=vn(x+yn) . Passing to a subsequence, we have

(3.15) ˆvnˆvE\{0},ˆvn(x)ˆv(x),a.e.xR3.

By (F2), (2.8), (3.6), and (3.15), we obtain

(3.16) m0+o(1)t3n=t3nΦ(un)=t3nΦ(t2n(vn)tn)=12vn22+14N[vn]13vn331t6nR3F(t2nvn)dx14R3F(t2nvn)t6ndx=14R3F(t2nˆvn)t6ndx,n.

This contradiction shows that {Q[un]} is bounded, and so {un} is bounded in E .

Since J(un)=0 , it follows from (1.7) and (1.9) that

(3.17) m0+o(1)=Φ(un)13J(un)=23R3[f(un)un3F(un)]dx.

We claim that there exist a δ>0 and a sequence ynR3 such that

(3.18) liminfnB1(yn)un3dx>δ.

Indeed, suppose that (3.18) does not hold. Then, we have

(3.19) limsupnsupyR3B1(y)un3dx=0.

By (F1) and Lemma 2.6, we have

(3.20) R3[f(un)un3F(un)]dx0.

This contradicts with (3.17) and Lemma 3.6 (ii). Therefore, (3.18) holds.

Let ˆun(x)=un(x+yn) . Then, we have ˆunE=unE and

(3.21) J(ˆun)=0,Φ(ˆun)m0,liminfnB1(0)ˆun3dx>δ.

Therefore, there exists ˆuE\{0} such that, passing to a subsequence,

(3.22) {ˆunˆu,inE;ˆunˆu,inLsloc(R3),s[3,6);ˆunˆu,a.e. onR3.

Let wn=ˆunˆu and

(3.23) Λ(u)23R3[f(u)u3F(u)]dx.

Then, (3.22) and Lemma 3.7 yield

(3.24) Φ(ˆun)=Φ(ˆu)+Φ(wn)+o(1)

and

(3.25) J(ˆun)=J(ˆu)+J(wn)+o(1).

From (1.7), (1.9), (3.21), (3.24), and (3.25), one has

(3.26) Λ(wn)=m0Λ(ˆu)+o(1)

and

(3.27) J(wn)=J(ˆu)+o(1).

If there exists a subsequence {wni} of {wn} such that wni=0 , then going to this subsequence, we have

(3.28) Φ(ˆu)=m0andJ(ˆu)=0,

which implies the conclusion of Lemma 3.8 holds. Next, we assume that wn0 . In view of Lemma 3.4, there exists tn>0 such that t2n(wn)tn . We claim that J(ˆu)0 . Otherwise, if J(ˆu)>0 , then (3.27) implies J(wn)<0 for large n . From (1.7), (1.9), (3.1), (3.5), and (3.26), we obtain

m0Λ(ˆu)+o(1)=Λ(wn)=Φ(wn)13J(wn)Φ(t2n(wn)tn)t3n3J(wn)m0t3n3J(wn)m0,

which is a contradiction due to Λ(ˆu)>0 ; thus J(ˆu)0 . Since ˆuE\{0} , in view of Lemma 3.4, there exists ˉt>0 such that ˉt2ˆuˉt . From (1.7), (1.9), (3.1), (3.5), (3.21), and Fatou’s lemma, one has

m0=limn[Φ(ˆun)13J(ˆun)]=limnΛ(un)Λ(ˆu)=Φ(ˆu)13J(ˆu)Φ(ˉt2ˆuˉt)ˉt33J(ˆu)m0ˉt33J(ˆu)m0,

which implies (3.28) also holds.□

Lemma 3.9

Assume that (F1), (F2), and (F3) hold. If ˆu and Φ(ˆu)=m0 , then ˆu is a critical point of Φ .

Proof

We prove this lemma by employing the method introduced in [8]. If the conclusion does not hold, we assume that Φ(ˆu)0 . Then, there exist δ>0 and ϱ>0 such that

(3.29) uˆuE3δΦ(u)ϱ.

Let {tn}R such that tn1 . Since t2nˆutnˆu in E , it follows from (2.11) and Lemma 2.5 that

(3.30) (t2nˆutn)ˆu22=R3(t2nˆutn)ˆu2dx=(t3n+1)R3ˆu2dx2R3(t2nˆutn)ˆudx=o(1)

and

(3.31) N(t2nˆutnˆu)=D((t2nˆutnˆu)2,(t2nˆutnˆu)2)=D((t2nˆutn)2,(t2nˆutn)2)+D(ˆu2,ˆu2)4D((t2nˆutn)2,(t2nˆutn)ˆu)4D(ˆu2,(t2nˆutn)ˆu)+4D((t2nˆutn)ˆu,(t2nˆutn)ˆu)+2D((t2nˆutn)2,ˆu2)=D((t2nˆutn)2,(t2nˆutn)2)D(ˆu2,ˆu2)+o(1)=(t3n1)D(ˆu2,ˆu2)+o(1)=o(1).

Combining (3.30) with (3.31), one has

(3.32) limt1t2ˆutˆuE=0.

Thus, there exists δ1>0 such that

(3.33) t1<δ1t2ˆutˆuE<δ.

In view of Lemma 3.2 and J(ˆu)=0 , one has

(3.34) Φ(t2ˆut)=Φ(ˆu)R3h(t,ˆu)dx=m0R3h(t,ˆu)dx,t>0.

It follows from (F1) and (1.9) that there exist T1(0,1) and T2(1,) such that

(3.35) J(T21ˆuT1)>0andJ(T22ˆuT2)<0.

Set Θmin{R3h(T1,ˆu)dx,R3h(T2,ˆu)dx} . Let SB(ˆu,δ) and εmin{Θ24,1,ϱδ8} . Then, [40, Lemma 2.3] yields a deformation ηC([0,1]×E,E) such that

  1. η(1,u)=u if Φ(u)<m02ε or Φ(u)>m0+2ε ;

  2. η(1,Φm0+εB(ˆu,δ))Φm0ε ;

  3. Φ(η(1,u))Φ(u),uE ;

  4. η(1,u) is a homeomorphism of E .

By Corollary 3.3, Φ(t2ˆut)Φ(ˆu)=m0 for t>0 , then it follows from (3.33) and (ii) that

(3.36) Φ(η(1,t2ˆut))m0ε,t>0,t1<δ1.

On the other hand, by (iii), (3.3), and (3.34), one has

(3.37) Φ(η(1,t2ˆut))Φ(t2ˆut)m0R3h(t,ˆu)dxm0δ2,t>0,t1δ1,

where

δ2min{R3h(1δ1,ˆu)dx,R3h(1+δ1,ˆu)dx}>0.

Combining (3.36) with (3.37), we have

(3.38) maxt[T1,T2]Φ(η(1,t2ˆut))<m0.

Define Ψ0(t)J(η(1,t2ˆut)) for t>0 . It follows from εΘ24 , (3.34) and (i) that η(1,t2ˆut)=t2ˆut for t=T1 and t=T2 , which, together with (3.35), implies

Ψ0(T1)=J(T21ˆuT1)>0andΨ0(T2)=J(T22ˆuT2)<0.

Since Ψ0(t) is continuous on (0,) , we have that η(1,t20ˆut0) for some t0[T1,T2] , which contradicts with (3.38). Thus, Φ(ˆu)=0 , and the proof is complete.□

Proof of Theorem 1.2

Applying Lemmas 3.6,3.8, and 3.9, we show Theorem 1.2.□

4 Existence of least energy solution

In this section, we give the proof of Theorem 1.3.

Proposition 4.1

[17] Let X be a Banach space, and let JR+ be an interval. We consider a family {Iλ}λD of C1 -functional on X of the form

Iλ(u)=A(u)λB(u),λD,

where B(u)0,uX , and such that either A(u)+ or B(u)+ , as u . We assume that there are two points v1,v2 in X such that

(4.1) cλinfγΓmaxt[0,1]Iλ(γ(t))>max{Iλ(v1),Iλ(v2)},

where

Γ={γC([0,1],X):γ(0)=v1,γ(1)=v2}.

Then, for almost every λD , there is a bounded (PS) cλ sequence for Iλ , i.e., there exists a sequence such that

  1. {un(λ)} is bounded in X ;

  2. Iλ(un(λ))cλ ;

  3. Iλ(un(λ))0 in X* , where X* is the dual of X.

To apply Proposition 4.1, we let X=E and introduce a family of functional defined by

(4.2) Φλ(u)=12R3u2dx+14R3R3u2(x)u2(y)4πxydxdyλR3[13u3+F(u)]dx,

for λ[12,1] .

Lemma 4.2

[16,29] Assume that (F1) and (F2) hold. Let u be a critical point of Φλ in H1(R3) , and then, we have the following Pohozaev-type identity:

(4.3) Pλ(u)12R3u2dx+54R3R3u2(x)u2(y)4πxydxdy3λR3[13u3+F(u)]dx=0.

We set Jλ(u)2Φλ(u),uPλ(u) , then

(4.4) Jλ(u)=32R3u2dx+34R3R3u2(x)u2(y)4πxydxdyλR3[u3+2f(u)u3F(u)]dx,

for λ[12,1] .

Choose wC0(R3) such that supp{w}>1 , and then, we have the following lemma.

Lemma 4.3

Assume that (F1) and (F2) hold. Then,

  1. there exists a T>0 independent of λ such that Φλ(T2wT)<0 for all λ[12,1] ;

  2. there exists a positive constant κ0 independent of λ such that for all λ[12,1] ,

    cλinfγΓmaxt[0,1]Φλ(γ(t))κ0>max{Φλ(0),Φλ(v2)},

    where

    Γ={γC([0,1],E):γ(0)=0,γ(1)=v2},andΦ12(v2)Φ12(T2wT)<0.

Proof

(i). It follows from (4.2) that

(4.5) Φλ(t2wt)=t32R3w2dx+t34R3R3w2(x)w2(y)4πxydxdyλ[t33R3w3dx+t3R3F(t2w)dx],t>0.

By (F2), one has that limt+[t6R3F(t2w)dx]=+ . Then, there exists T>0 such that

(4.6) Φ12(T2wT)=T32R3w2dx+T34R3R3w2(x)w2(y)4πxydxdy12[T33R3w3dx+T3R3F(T2w)dx]<0.

Hence,

(4.7) Φλ(T2wT)=T32R3w2dx+T34R3R3w2(x)w2(y)4πxydxdyλ2[T33R3w3dx+T3R3F(T2w)dx]<0,λ[12,1].

(ii). By (F1), (2.2), (2.4), (2.8), and (4.2), we have

(4.8) Φλ(u)=12R3u2dx+14R3R3u2(x)u2(y)4πxydxdyλR3[13u3+F(u)]dx12u22+14N[u]13u33R3F(u)dx12u22+14N[u]25u33C1u6614u22+9100N[u]C1u669100Q[u]C2(Q[u])3,λ[12,1],

which yields that there exists δ>0 and κ0>0 such that for all λ[12,1] ,

(4.9) Φλ(u)>0,u{vE:0<Q[u]<δ},Φλ(u)κ0,u{vE:Q[u]=δ}.

It follows from (4.7) and (4.9) that (ii) holds.□

Lemma 4.4

Assume that (F1), (F2), and (F4) hold. Then, for almost every λ[12,1] , there exists a uλE\{0} such that

(4.10) Φλ(uλ)=0andΦλ(uλ)=cλκ0.

Proof

In view of (F1), (F3), and Lemma 4.3, we obtain that Iλ(u) satisfies the assumptions of Proposition 4.1 with X=E and Iλ=Φλ . So for almost every λ[12,1] , there exists a bounded sequence {un(λ)}H1(R3) (for simplicity, we denote {un} instead of {un(λ)} such that

(4.11) Φλ(un)cλ>0,Φλ(un)0.

Rest of the proof is standard (cf. [16, Proposition 3.6, Corollary 3.7]), so we omit it.□

Proof of Theorem 1.3

In view of Lemma 4.4, there exist two sequences of {λn}[12,1] and {uλn}E , denoted by {un} , such that

(4.12) λn1,Φ(un)=0,κ0Φλn(un)=cλn.

From (4.2), (4.4), and (4.12), one has

(4.13) c12cλn=Φλn(un)13Jλn(un)=2λn3R3[f(un)un3F(un)]dx.

We claim that {un} is bounded in E . Otherwise, we may suppose that Q[un] . Set tn3Q[un] and vn(x)t2nun(xtn) , then Q[vn]=1 . By (F1) and (F2), we can assume that

(4.14) f(t)t218C3f(t)tR0,

where C3 is the embedding constant, and R0 is given in (F4). Set κ=κ(κ1) and

(4.15) Ω1n{xR3:f(un)unun18C3},Ω2n{xR3:f(un)unR0}.

Then, (F4), (4.14), and (4.15) imply that κ[32,3) and Ω1nΩ2n . By (F1) and (F2), there exist 0<t1<t2<+ such that

(4.16) f(t)tt18C3,tt1andf(t)t>R0,tt2.

Hence, it follows from (F1) and (F4) that there exists C1>0 such that

(4.17) f(t)tκC1[f(t)t3F(t)],t1tt2.

By (2.8), (4.15), and (4.16), one has

(4.18) 1Q[un]Ω1nf(un)ununun3dx18C3Q[un]un33=18C3vn3318.

From (2.8), (4.12), (4.13), (4.15), (4.16), (4.17) and the Hölder inequality, we have

(4.19) 1Q[un]Ω2n\Ω1nf(un)unun2dx1Q[un][Ω2n\Ω1nf(un)unκdx]1κun22κC1κ1Q[un](Ω2n\Ω1n[f(un)un3F(un)]dx)1κun22κC2Q[un]un22κ=C2(Q[un])3κ3κvn22κ=o(1).

By virtue of (F4), (2.8), (4.12), (4.13), (4.15), and the Hölder inequality, one can obtain

(4.20) 1Q[un]R3\Ω2nf(un)unun2dx1Q[un][R3\Ω2nf(un)unκdx]1κun22κC3Q[un](R3\Ω2n[f(un)un3F(un)]dx)1κun22κC4Q[un]un22κ=C4(Q[un])3κ3κvn22κ=o(1).

Hence, it follows from (2.2), (2.4), (4.2), (4.12), (4.18) (4.19), and (4.20) that

1412Q[un](un22+N[un])1Q[un](un22+N[un]λnun33)=λnQ[un]R3f(un)undx1Q[un][Ω1nf(un)undx+Ω2n\Ω1nf(un)undx+R3\Ω2nf(un)undx]=1Q[un][Ω1nf(un)ununun3dx+Ω2n\Ω1nf(un)unun2dx+R3\Ω2nf(un)unun2dx]18+o(1).

This contradiction shows that supnNQ[un]< , i.e., {un} is bounded in E .

By (F1), (4.12), (4.13), and Lemma 2.6, it is easy to prove that δ1limsupnsupyR3B1(y)un3dx>0 . Going if necessary to a subsequence, we may assume the existence of ynR3 such that B1(yn)un3dx>δ12 . Let ˜un(x)=un(x+yn) . Then, ˜unE=unE=1 , and

(4.21) B1(0)˜un3dx>δ12.

Passing to a subsequence, we have ˜un˜u in E , ˜un˜u in Lsloc(R3) , 3s<6 , ˜un˜u a.e. on R3 . Obviously, (4.21) implies that ˜u0 . By a standard argument, we can prove that Φ(˜u)=0 and Φ(˜u)=c1=infγΓmaxt[0,1]Φ(γ(t)) ; these yield that ˜u is a least energy solution of problem (1.1) by noting that for any nontrivial solution uE\{0} , there holds Φ(u)c1 . (cf. [16, Proposition 3.4]).□


,

Acknowledgements

The authors would like to thank the anonymous referees for their invaluable suggestions.

  1. Funding information: This work was supported by the National Natural Science Foundation of China (No. 12171486), the Project of Key Laboratory of Language Engineering and Computing of Guangdong Province (No. LEC2020ZBKT002), the Scientific Research Fund of Hunan Provincial Education Department, China (22B1103), the Science and Technology Innovation Program of Hunan Province (No. 2024RC3021), the Young Backbone Teachers Project of Hunan Province, and Natural Science Foundation for Excellent Young Scholars of Hunan Province (No. 2023JJ20057).

  2. Author contributions: All authors contributed equally to the writing of this article. All authors read and approved the final manuscript.

  3. Conflict of interest: The authors declare that they have no competing interests.

  4. Data availability statement: Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.

References

[1] A. Ambrosetti and D. Ruiz, Multiple bound-states for the Schrödinger-Poisson problem, Commun. Contemp. Math. 10 (2008), 391–404, https://doi.org/10.1142/S021919970800282X. Search in Google Scholar

[2] A. Azzollini and A. Pomponio, Ground state solutions for the nonlinear Schrödinger-Maxwell equations, J. Math. Anal. Appl. 345 (2008), 90–108, https://doi.org/10.1016/j.jmaa.2008.03.057. Search in Google Scholar

[3] V. Benci and D. Fortunato, An eigenvalue problem for the Schrödinger-Maxwell equations, Topol. Methods Nonlinear Anal. 11 (1998), 283–293, https://doi.org/10.12775/TMNA.1998.019. Search in Google Scholar

[4] H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. I. Existence of a ground-state, Arch. Rational Mech. Anal. 82 (1983), 313–345, https://doi.org/10.1007/BF00250555. Search in Google Scholar

[5] G. Cerami and G. Vaira, Positive solutions for some non-autonomous Schrödinger-Poisson systems, J. Differential Equations 248 (2010), 521–543, https://doi.org/10.1016/j.jde.2009.06.017. Search in Google Scholar

[6] S. T. Chen, A. Fiscella, P. Pucci, and X. H. Tang, Semiclassical ground-state solutions for critical Schrödinger-Poisson systems with lower perturbations, J. Differential Equations 268 (2020), 2672–2716, https://doi.org/10.1016/j.jde.2019.09.041. Search in Google Scholar

[7] S. T. Chen and X. H. Tang, Axially symmetric solutions for the planar Schrödinger-Poisson system with critical exponential growth, J. Differential Equations 269 (2020), 9144–9174, https://doi.org/10.1016/j.jde.2020.06.043. Search in Google Scholar

[8] S. T. Chen and X. H. Tang, Berestycki-Lions conditions on ground-state solutions for a nonlinear Schrödinger equation with variable potentials, Adv. Nonlinear Anal. 9 (2020), 496–515, https://doi.org/10.1515/anona-2020-0011. Search in Google Scholar

[9] Z. Chen, D. D. Qin, and W. Zhang, Localized nodal solutions of higher topological type for nonlinear Schrödinger-Poisson system, Nonlinear Anal. 198 (2020), 111896, https://doi.org/10.1016/j.na.2020.111896. Search in Google Scholar

[10] G. M. Coclite, A multiplicity result for the nonlinear Schrödinger-Maxwell equations, Commun. Appl. Anal. 7 (2003), 417–423. Search in Google Scholar

[11] T. D’Aprile and D. Mugnai, Non-existence results for the coupled Klein-Gordon-Maxwell equations, Adv. Nonlinear Stud. 4 (2004), 307–322, https://doi.org/10.1515/ans-2004-0305. Search in Google Scholar

[12] X. L. Dou, X. M. He, and V. D. Rădulescu, Multiplicity of positive solutions for the fractional Schrödinger-Poisson system with critical nonlocal term, Bull. Math. Sci. 14 (2024), 2350012, https://doi.org/10.1142/S1664360723500121. Search in Google Scholar

[13] S. H. Feng, J. H. Chen, and X. J. Huang, Critical fractional Schrödinger-Poisson systems with lower perturbations: the existence and concentration behavior of ground-state solutions, Adv. Nonlinear Anal. 13 (2004), 20240006, https://doi.org/10.1515/anona-2024-0006. Search in Google Scholar

[14] Y. Gu and F. F. Liao, Ground state solutions of Nehari-Pohozaev type for Schrödinger-Poisson-Slater equation with zero mass and critical growth, J. Geom. Anal. 34 (2024), 221, https://doi.org/10.1007/s12220-024-01656-z. Search in Google Scholar

[15] L. R. Huang, E. M. Rocha, and J. Q. Chen, Two positive solutions of a class of Schrödinger-Poisson system with indefinite nonlinearity, J. Differential Equations 255 (2013), 2463–2483, https://doi.org/10.1016/j.jde.2013.06.022. Search in Google Scholar

[16] I. Ianni and D. Ruiz, Ground and bound-states for a static Schrödinger-Poisson-Slater problem, Commun. Contemp. Math. 14 (2012), no. 1, 1250003, https://doi.org/10.1142/S0219199712500034. Search in Google Scholar

[17] L. Jeanjean, On the existence of bounded Palais-Smale sequence and application to a Landesman-Lazer type problem set on RN, Proc. Roy. Soc. Edinburgh Sect. A 129 (1999), 787–809, https://doi.org/10.1017/S0308210500013147. Search in Google Scholar

[18] L. Jeanjean and J. Toland, Bounded Palais-Smale mountain-pass sequences, C. R. Acad. Sci. Paris Sér. I Math. 327 (1998), 23–28, https://doi.org/10.1016/S0764-4442(98)80097-9. Search in Google Scholar

[19] C. Y. Lei, J. Lei, and H. M. Suo, Ground state for the Schrödinger-Poisson-Slater equation involving the Coulomb-Sobolev critical exponent, Adv. Nonlinear Anal. 12 (2023), 20220299, https://doi.org/10.1515/anona-2022-0299. Search in Google Scholar

[20] C. Y. Lei and Y. T. Lei, On the existence of ground-states of an equation of Schrödinger-Poisson-Slater type, C. R. Math. Acad. Sci. Paris, 359 (2021), 219–227, https://doi.org/10.5802/crmath.175. Search in Google Scholar

[21] C. Y. Lei, V. D. Rădulescu, and B. L. Zhang, Ground states of the Schrödinger-Poisson-Slater equation with critical growth, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 117 (2023), no. 3, 128, https://doi.org/10.1007/s13398-023-01457-z. Search in Google Scholar

[22] A. R. Li, C. Q. Wei, and L. G. Zhao, Existence of nontrivial distributional solutions for a class of Schrödinger-Poisson system with Sobolev critical nonlinearity and zero mass, J. Math. Phys. 64 (2023), 121510, https://doi.org/10.1063/5.0141514. Search in Google Scholar

[23] Y. Q. Li, T. V. Nguyen, and B. L. Zhang, Existence, concentration and multiplicity of solutions for (p,N)-Laplacian equations with convolution term, Bull. Math. Sci. (2024), 2450009, https://doi.org/10.1142/S1664360724500097. Search in Google Scholar

[24] F. F. Liao, D. D. Qin, X. H. Tang, and J. Y. Wei, Multiple solutions for critical Schrödinger-Poisson-Slater equation with lower order perturbation and zero mass, Preprint. Search in Google Scholar

[25] E. H. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. Math. 118 (1983), no. 2, 349–374, https://doi.org/10.2307/2007032. Search in Google Scholar

[26] E. H. Lieb and M. Loss, Analysis, Graduate Studies in Mathematics, vol. 14, American Mathematical Society, Providence, 2001. 10.1090/gsm/014Search in Google Scholar

[27] P. L. Lions, Solutions of Hartree-Fock equations for Coulomb systems, Comm. Math. Phys. 109 (1984), 33–97, https://doi.org/10.1007/BF01205672. Search in Google Scholar

[28] Z. S. Liu, V. D. Rădulescu, C. L. Tang, and J. J. Zhang, Another look at planar Schrödinger-Newton systems, J. Differential Equations 328 (2022), 65–104, https://doi.org/10.1016/j.jde.2022.04.035. Search in Google Scholar

[29] Z. S. Liu, Z. T. Zhang, and S. B. Huang, Existence and nonexistence of positive solutions for a static Schrödinger-Poisson-Slater equation, J. Differential Equations 266 (2019), 5912–5941, https://doi.org/10.1016/j.jde.2018.10.048. Search in Google Scholar

[30] N. J. Mauser, The Schrödinger-Poisson-Xα equation, Appl. Math. Lett. 14 (2001), 759–763, https://doi.org/10.1016/S0893-9659(01)80038-0. Search in Google Scholar

[31] C. Mercuri, V. Moroz, and J. Van Schaftingen, Groundstates and radial solutions to nonlinear Schrödinger-Poisson-Slater equations at the critical frequency, Calc. Var. Partial Differential Equations 55 (2016), 146, https://doi.org/10.1007/s00526-016-1079-3. Search in Google Scholar

[32] D. Ruiz, On the Schrödinger-Poisson-Slater system: behavior of minimizers, radial and nonradial cases, Arch. Ration. Mech. Anal. 198 (2010), 349–368, https://doi.org/10.1007/s00205-010-0299-5. Search in Google Scholar

[33] D. Ruiz, The Schrödinger-Poisson equation under the effect of a nonlinear local term, J. Funct. Anal. 237 (2006), 655–674, https://doi.org/10.1016/j.jfa.2006.04.005. Search in Google Scholar

[34] J. Seok, On nonlinear Schrödinger-Poisson equations with general potentials, J. Math. Anal. Appl. 401 (2013), 672–681, https://doi.org/10.1016/j.jmaa.2012.12.054. Search in Google Scholar

[35] J. Slater, A simplification of the Hartree-Fock method, Phys. Rev. 81 (1951), 385–390, https://doi.org/10.1103/PhysRev.81.385. Search in Google Scholar

[36] J. J. Sun and S. W. Ma, Ground state solutions for some Schrödinger-Poisson systems with periodic potentials, J. Differential Equations 260 (2016), 2119–2149, https://doi.org/10.1016/j.jde.2015.09.057. Search in Google Scholar

[37] X. H. Tang and S. T. Chen, Ground state solutions of Nehari-Pohozaev type for Schrödinger-Poisson problems with general potentials, Discrete Contin. Dyn. Syst. 37 (2017), 4973–5002, https://doi.org/10.3934/dcds.2017214. Search in Google Scholar

[38] X. P. Wang and F. F. Liao, Existence and nonexistence of solutions for Schrödinger-Poisson problems, J. Geom. Anal. 33 (2023), no. 2, 56, https://doi.org/10.1007/s12220-022-01104-w. Search in Google Scholar

[39] L. X. Wen, S. T. Chen, and V. D. Rădulescu, Axially symmetric solutions of the Schrödinger-Poisson system with zero mass potential in R2, Appl. Math. Lett. 104 (2020), 106244, https://doi.org/10.1016/j.aml.2020.106244. Search in Google Scholar

[40] M. Willem, Minimax Theorems, Birkhäuser, Boston, 1996, https://doi.org/10.1007/978-1-4612-4146-1. Search in Google Scholar

[41] L. Yang and Z. S. Liu, Infinitely many solutions for a zero mass Schrödinger-Poisson-Slater problem with critical growth, J. Appl. Anal. Comput. 5 (2019), 1706–1718, https://doi.org/10.11948/20180273. Search in Google Scholar

[42] L. G. Zhao and F. K. Zhao, On the existence of solutions for the Schrödinger-Poisson equations, J. Math. Anal. Appl. 346 (2008), 155–169, https://doi.org/10.1016/j.jmaa.2008.04.053. Search in Google Scholar

[43] L. G. Zhao and F. K. Zhao, Positive solutions for Schrödinger-Poisson equations with a critical exponent, Nonlinear Anal. 70 (2009), 2150–2164, https://doi.org/10.1016/j.na.2008.02.116. Search in Google Scholar

[44] T. T. Zheng, C. Y. Lei, and J. F. Liao, Multiple positive solutions for a class of concave-convex Schrödinger-Poisson-Slater equations with critical exponent, Adv. Nonlinear Anal. 13 (2024), 20230129, https://doi.org/10.1515/anona-2023-0129. Search in Google Scholar

[45] T. T. Zheng, C. Y. Lei, and J. F. Liao, Multiple positive solutions for a Schrödinger-Poisson-Slater equation with critical growth, J. Math. Anal. Appl. 525 (2023), 127206, https://doi.org/10.1016/j.jmaa.2023.127206. Search in Google Scholar

Received: 2024-11-07
Revised: 2025-01-08
Accepted: 2025-02-20
Published Online: 2025-03-13

© 2025 the author(s), published by De Gruyter

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

Articles in the same Issue

  1. Research Articles
  2. Incompressible limit for the compressible viscoelastic fluids in critical space
  3. Concentrating solutions for double critical fractional Schrödinger-Poisson system with p-Laplacian in ℝ3
  4. Intervals of bifurcation points for semilinear elliptic problems
  5. On multiplicity of solutions to nonlinear Dirac equation with local super-quadratic growth
  6. Entire radial bounded solutions for Leray-Lions equations of (p, q)-type
  7. Combinatorial pth Calabi flows for total geodesic curvatures in spherical background geometry
  8. Sharp upper bounds for the capacity in the hyperbolic and Euclidean spaces
  9. Positive solutions for asymptotically linear Schrödinger equation on hyperbolic space
  10. Existence and non-degeneracy of the normalized spike solutions to the fractional Schrödinger equations
  11. Existence results for non-coercive problems
  12. Ground states for Schrödinger-Poisson system with zero mass and the Coulomb critical exponent
  13. Geometric characterization of generalized Hajłasz-Sobolev embedding domains
  14. Subharmonic solutions of first-order Hamiltonian systems
  15. Sharp asymptotic expansions of entire large solutions to a class of k-Hessian equations with weights
  16. Stability of pyramidal traveling fronts in time-periodic reaction–diffusion equations with degenerate monostable and ignition nonlinearities
  17. Ground-state solutions for fractional Kirchhoff-Choquard equations with critical growth
  18. Existence results of variable exponent double-phase multivalued elliptic inequalities with logarithmic perturbation and convections
  19. Homoclinic solutions in periodic partial difference equations
  20. Classical solution for compressible Navier-Stokes-Korteweg equations with zero sound speed
  21. Properties of minimizers for L2-subcritical Kirchhoff energy functionals
  22. Asymptotic behavior of global mild solutions to the Keller-Segel-Navier-Stokes system in Lorentz spaces
  23. Blow-up solutions to the Hartree-Fock type Schrödinger equation with the critical rotational speed
  24. Qualitative properties of solutions for dual fractional parabolic equations involving nonlocal Monge-Ampère operator
  25. Regularity for double-phase functionals with nearly linear growth and two modulating coefficients
  26. Uniform boundedness and compactness for the commutator of an extension of Riesz transform on stratified Lie groups
Downloaded on 18.7.2025 from https://www.degruyterbrill.com/document/doi/10.1515/anona-2025-0073/html
Scroll to top button