# An Existence Result for a Generalized Quasilinear Schrödinger Equation with Nonlocal Term

Authors: Quanqing Li; Kaimin Teng; Jian Zhang; Jianjun Nie
Journal: Journal of Function Spaces
 (2020)

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## Abstract

In this paper, we consider the following generalized quasilinear Schrödinger equation with nonlocal term−divg2u∇u+gug′u∇u2+Vxu=λx−μ∗upup−2u,x∈ℝN, where N≥3, g:ℝ→ℝ+ is a C1 even function, g0=1, g′s≥0 is for all s≥0, lim∣s∣→+∞gs/sα−1≔β>0 is for some α>1, and α−1gs≥g′ss is for all s≥0, 2α≤p≤2αN−μ/N−2, and 0<μ<N. We prove that the equation admits a solution by using a constrained minimization argument.


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## Body

## 1. Introduction and Preliminaries

The main purpose of this paper is to investigate the existence of solutions for the following generalized quasilinear Schrödinger equation with nonlocal term(1)−divg2u∇u+gug′u∇u2+Vxu=λx−μ∗upup−2u,x∈ℝN,where N≥3, g:ℝ→ℝ+ is a C1 even function, g0=1, g′s≥0 is for all s≥0, lim∣s∣→+∞gs/sα−1≔β>0 is for some α>1, and α−1gs≥g′ss is for all s≥0, 2α≤p≤2αN−μ/N−2, and 0<μ<N.Whengu=1, (1) boils down to the socalled nonlinear Choquard or Choquard-Pekar equation(2)−Δu+Vxu=λx−μ∗upup−2u,x∈ℝN.Such like equation has several physical origins. The problem(3)−Δu+u=x−1∗u2u,x∈ℝ3appeared at least as early as in 1954, in a work by Pekar describing the quantum mechanics of a polaron at rest [1]. In 1976, Choquard used (3) to describe an electron trapped in its own hole and in a certain approximation to Hartree-Fock theory of one component plasma [2]. In 1996, Penrose proposed (3) as a model of self-gravitating matter, in a program in which quantum state reduction is understood as a gravitational phenomenon [3]. In this context, equation of type (3) is usually called the nonlinear Schrödinger-Newton equation. The first investigations for existence and symmetry of the solutions to (3) go back to the works of Lieb [2] and Lions [4]. In [2], by using symmetric decreasing rearrangement inequalities, Lieb proved that the ground state solution of equation (3) is radial and unique up to translations. Lions [4] showed the existence of a sequence of radially symmetric solutions. Since then, many efforts have been made to study the existence of nontrivial solutions for nonlinear Choquard equations. Wei and Winter [5] showed that the ground state solution is nondegenerate. Ma and Zhao [6] considered the generalized Choquard equation(4)−Δu+u=x−μ∗uquq−2uq≥2and proved that every positive solution of it is radially symmetric and monotone decreasing about some fixed point, under the assumption that a certain set of real numbers, defined in terms of N,μ, and q, is nonempty. Under the same assumption, Cingolani, Clapp, and Secchi [7] gave some existence and multiplicity results in the electromagnetic case and established the regularity and some decay asymptotically at infinity of the ground states. In [8], Moroz and Van Schaftingen eliminated this restriction and showed the regularity, positivity, and radial symmetry of the ground states for the optimal range of parameters and derived decay asymptotically at infinity for them as well. Moreover, they [9] also obtained a similar conclusion under the assumption of Berestycki-Lions type nonlinearity. We point out that the existence, multiplicity, and concentration of such like equation have been established by many authors. We refer the readers to [10, 11] for the existence of sign-changing solutions, [5, 12] for the existence and concentration behavior of the semiclassical solutions and [13] for the critical nonlocal part with respect to the Hardy-Littlewood-Sobolev inequality. For more details associated with the Choquard equation, please refer to [14–16] and the references therein.In the past, even the research on the existence of solitary wave solutions for the Schrödinger equation with local term(5)−divg2u∇u+gug′u∇u2+Vxu=fx,u,x∈ℝNis for some given special function g·, see [17–19]. However, related to the nonlocal equation (1), as far as we know, there is no result in this direction. In this paper, with the aid of the new variable replacement developed by Shen and Wang in [18] and inspired by [20, 21], existence of solutions for equation (1) have been established. Problem (1) has a variational structure, and the corresponding energy functional is defined by(6)Iu=12∫ℝNg2u∇u2dx+12∫ℝNVxu2dx−λ2p∫ℝNx−μ∗upupdx=12∫ℝNg2u∇u2dx+12∫ℝNVxu2dx−λ2p∫ℝ2Nuxpuypx−yμdxdy.However,I is not well defined in H1ℝN because of the term ∫ℝNg2u∇u2dx. To overcome this difficulty, we make a change of variable constructed by Shen and Wang in [18]: v≔Gu≔∫0ugtdt. Then, we obtain(7)Jv=12∫ℝN∇v2dx+12∫ℝNVxG−1v2dx−λ2p∫ℝNx−μ∗G−1vpG−1vpdx=12∫ℝN∇v2dx+12∫ℝNVxG−1v2dx−λ2p∫ℝ2NG−1vxpG−1vypx−yμdxdy.We say thatu is a weak solution of (1), if(8)I′u,φ=∫ℝNg2u∇u∇φ+gug′u∇u2φ+Vxuφ−λx−μ∗upup−2uφdx=0for all φ∈C0∞ℝN. Let φ=1/guψ. By [18], we know that the above formula is equivalent to(9)J′v,ψ=∫ℝN∇v∇ψ+VxG−1vgG−1vψ−λx−μ∗G−1vpG−1vp−2G−1vψgG−1vdx=0for all ψ∈C0∞ℝN. Therefore, in order to find the solution of (1), it suffices to study the solution of the following equation:(10)−Δv+VxG−1vgG−1v−λx−μ∗G−1vpG−1vp−2G−1vgG−1v=0.In this paper, we assume that the following condition holds.VV∈CℝN,ℝ, 0<V0≔infx∈ℝNVx, and lim∣x∣→∞Vx=+∞.SetHV1ℝN=v∈H1ℝN:∫ℝN∇v2+Vxv2dx<+∞ with the norm(11)∥v∥HV12=∫ℝN∇v2+Vxv2dx.Then, by the proof of Lemma 4 in [22], the embedding HV1ℝN°LtℝN is compact for all t∈2,2∗. Moreover, for any a>0, we define ma≔infv∈MaEv, where(12)Ma≔v∈HV1ℝN:∫ℝ2NG−1vxpG−1vypx−yμdxdy=aEv≔∫ℝN∇v2+VxG−1v2dx.Our main result is the following:Theorem 1.
Suppose thatV is satisfied, then, there exists λn→+∞ such that equation (1) with λ=λn has a solution.


## 2. Proof of Theorem 1

To begin with, we give some lemmas.Lemma 2 (see [23, 24]).
The functionsg,G and G−1 possess the following properties:(1)
Gs≤gss≤αGs for all s≥0; Gs≥gss≥αGs for all s≤0(2)
G−1ss/gG−1s≤G−1s2≤αG−1ss/gG−1s for all s∈ℝ(3)
sα≤α/β∣Gs∣ for all s∈ℝProposition 3 [25] (Hardy-Littlewood-Sobolev inequality).
Letr,t>1 and 0<μ<N with 1/r+μ/N+1/t=2. Let g∈LrℝN and h∈LtℝN. Then, there exists a sharp constant Cr,N,μ,t independent of g and h such that(13)∣∫ℝN∫ℝNgxhyx−yμdxdy∣≤Cr,N,μ,t∥g∥r∥h∥t.Proof of Theorem 1.
The proof consists of two steps.
Step 1: we prove that for eacha>0, ma is achieved at some va∈Ma, which is a weak solution of equation (10) with λ=λa satisfying λa∈ma/aα,αma/a.
For fixeda>0, let vn⊂Ma be a minimizing sequence for ma, i.e., vn∈HV1ℝN satisfying ∫ℝ2NG−1vnxpG−1vnyp/x−yμdxdy=a such that Evn→ma as n→∞. We assert that there exists a constants C1>0 such that Evn≥C1∫ℝN∇vn2+Vxvn2dx. Indeed, we may assume that vn≠0 (otherwise, the conclusion is trivial). If the conclusion is false, then for any positive integer n, we may assume that(14)Evn=∫ℝN∇vn2+VxG−1vn2dx<1n∥vn∥HV12=1n∫ℝN∇vn2+Vxvn2dx.
Setwn=vn/∥vn∥HV1 and gn=G−1vn2/∥vn∥HV12. Then,(15)∫ℝN∇wn2dx+∫ℝNVxgnxdx→0,which implies that(16)∫ℝN∇wn2dx→0,∫ℝNVxgnxdx→0,∫ℝNVxwn2dx→1as n→∞. Then for each ε>0, there exists a constant C2>0 independent of n such that measΩn<ε, where Ωn≔x∈ℝN:∣G−1vnx∣≥C2. Otherwise, there exist ε0>0 and a subsequence G−1vnk of G−1vn such that for any positive integer k,(17)measΩnk≥ε0>0,where Ωnk=x∈ℝN:G−1vnkx≥k. By V, one has(18)C≥Evnk≥∫ℝNVxG−1vnkx2dx≥∫ΩnkVxG−1vnkx2dx≥V0k2ε0→+∞as k→+∞, a contradiction. Noting that as ∣G−1vn∣<C2, by Lemma 2 (1) and monotonicity of g, we have(19)vn2≤g2G−1vnG−1vn2≤g2C2G−1vn2.
Hence,(20)∫ℝN\ΩnVxwn2dx≤g2C2∫ℝNVxgnxdx→0.
By the integral absolutely continuity, there existsε>0 such that whenever Ω⊂ℝN and measΩ<ε, ∫ΩVxwn2dx<1/2. For this ε, one has(21)∫ℝNVxwn2dx=∫ΩnVxwn2dx+∫ℝN\ΩnVxwn2dx≤12+∫ℝNΩnVxwn2dx,which implies 1≤1/2, a contradiction. Therefore, up to a subsequence, there exists va∈HV1ℝN such that vn→va in HV1ℝN, vn→va in LtℝN for 2≤t<2∗, and vnx→vax, a.e., on ℝN. By means of the definition of weak convergence, we know(22)0≤∫ℝN∇vn−∇va2dx=∫ℝN∇vn2dx+∫ℝN∇va2dx−2∫ℝN∇vn·∇vadx,which implies that(23)liminfn→∞∫ℝN∇vn2dx≥liminfn→∞2∫ℝN∇vn·∇vadx−∫ℝN∇va2dx=∫ℝN∇va2dx.
By Fatou Lemma, we have(24)∫ℝNVxG−1va2dx≤liminfn→∞∫ℝNVxG−1vn2dx.
Consequently,Eva≤liminfn→∞∫ℝN∇vn2+∫ℝNVxG−1vn2=liminfn→∞Evn. Moreover, by the Hardy-Littlewood-Sobolev inequality and Lemma 2 (3), one has(25)a=∫ℝ2NG−1vnxpG−1vnypx−yμdxdy≤C∥G−1vnp∥2N/2N−μ2≤C∫ℝNvn2Np/α2N−μdx2N−μ/N.
Since2<2Np/α2N−μ<2∗, by Lemma A.1 in [26] and Lebesgue’s dominated convergence theorem, we can easily infer that ∫ℝ2NG−1vaxpG−1vayp/x−yμdxdy=a, and so va∈Ma. Hence, ma≤Eva≤liminfn→∞Evn=ma, which means that ma is achieved at some va∈Ma. Moreover, by a standard argument, we can conclude that va is a weak solution of(26)−Δv+VxG−1vgG−1v=λax−μ∗G−1vpG−1vp−2G−1vgG−1v.
Multiplying the above equation byva and integrating over ℝN, one has(27)∫ℝN∇va2dx+∫ℝNVxG−1vavagG−1vadx=λa∫ℝNx−μ∗G−1vapG−1vap−2G−1vavagG−1vadx.
By Lemma 2 (2), we obtainma/aα≤λa≤αma/a. Indeed, by Lemma 2 (2), we have(28)1αma=1αEva=1α∫ℝN∇va2dx+∫ℝNVxG−1va2dx≤∫ℝN∇va2dx+1α∫ℝNVxG−1va2dx≤∫ℝN∇va2dx+∫ℝNVxG−1vavagG−1vadx≤λa∫ℝNx−μ∗G−1vapG−1vapdx=λa·a,
i.e.,λa≥ma/aα. Furthermore,(29)ma=Eva=∫ℝN∇va2dx+∫ℝNVxG−1va2dx≥∫ℝN∇va2dx+∫ℝNVxG−1vavagG−1vadx≥λaα∫ℝNx−μ∗G−1vapG−1vapdx=λaα·a,i.e., λa≤αma/a.
Step 2: we prove thatλa→+∞ as a→0.
If the conclusion is false, then there exists a constantG0>0 and an→0n→∞ such that λn≔λan≤G0. Set vn≔van, by Lemma 2 (2) and Hardy-Littlewood-Sobolev inequality, we have(30)∫ℝN∇vn2dx+∫ℝNVxG−1vnvngG−1vndx=λn∫ℝNx−μ∗G−1vnpG−1vnp−2G−1vnvngG−1vndx≤λn∫ℝNx−μ∗G−1vnpG−1vnpdx=λnan≤G0·an→0as n→∞. Since 2<2Np/α2N−μ<2∗, there exists a constant θ∈0,1 such that 1/2Np/α2N−μ=θ/2+1−θ/2∗. Consequently, by Lemma 2 (3), V, Hölder inequality, and Young inequality, one has(31)∫ℝNG−1vn2Np/2N−μdx≤C∫ℝNvn2Np/α2N−μdx=C∫ℝNvnθ2Np/α2N−μvn1−θ2Np/α2N−μdx≤C∥vn∥2θ2Np/α2N−μ∥vn∥2∗1−θ2Np/α2N−μ≤Cθ∥vn∥22Np/α2N−μ+C1−θ∥vn∥2∗2Np/α2N−μ≤C∫ℝNVxvn2dxNp/α2N−μ+C∫ℝN∇vn2dxNp/α2N−μ≤C∫ℝN∇vn2+Vxvn2dxNp/α2N−μ≤C∫ℝN∇vn2+VxG−1vn2dxNp/α2N−μ.
Hence, again, by Lemma 2 (2) and Hardy-Littlewood-Sobolev inequality, we have(32)∫ℝN∇vn2dx+∫ℝNVxG−1vnvngG−1vndx≤λn∫ℝNx−μ∗G−1vnpG−1vnpdx≤Cλn∥G−1vnp∥2N/2N−μ2≤C·G0∫ℝN∇vn2+VxG−1vn2dxp/α≤C·G0∫ℝN∇vn2+VxG−1vnvngG−1vndxp/α,and so ∫ℝN∇vn2+VxG−1vnvn/gG−1vndx≥C since p/α≥2>1, a contradiction. By steps 1 and 2, we complete the proof of Theorem 1.



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