Existence and multiplicity results for some Schrödinger-Poisson system with critical growth

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Abstract

In this paper we study the existence and multiplicity of positive solutions for the Schrödinger-Poisson system with critical growth:{ε2Δu+V(x)u=f(u)+|u|3uϕ,xR3,ε2Δϕ=|u|5,xR3,uH1(R3),u(x)>0,xR3, where ε>0 is a parameter, V:R3R is a continuous function and f:RR is a C1 function. Under a global condition for V we prove that the above problem has a ground state solution and relate the number of positive solutions with the topology of the set where V attains its minimum, by using variational methods.

Introduction

In this paper we are concerned with the following Schrödinger-Poisson system involving critical growth{ε2Δu+V(x)u=f(u)+|u|3uϕ,xR3,ε2Δϕ=|u|5,xR3,uH1(R3),u(x)>0,xR3, where ε>0 is a parameter, V and f are satisfied some suitable conditions which will be stated below.

The investigation of equation (1.1) is motivated by recent studies of Schrödinger-Poisson system{Δu+bu+λϕg(u)=f(u),xR3,Δϕ=2G(u),xR3, where the functions g(u) and G(u) satisfy |g(u)|C(|u|+|u|q) for some q[1,4),G(u)=0ug(t)dt, and f(u) satisfies |f(u)|C(|u|+|u|p) for some p(1,5]. Eq. (1.2) arises in many interesting mathematical physics contexts, such as in quantum electro-dynamics, to describe the interaction between a charge particle interacting with electromagnetic field, and also in semi-conductor theory, in nonlinear optics and in plasma physics. We refer to [7], [4], [12], [34] for more details on physical aspects.

For subcritical nonlinearity f with p(1,5) and subcritical nonlocal term g with q[1,4), problem (1.2) was studied by several authors, see for instance, [8], [28]. In [8], system (1.2) on bounded domain ΩR3 was considered for positive and negative value of λ. In [28] system (1.2) was studied and it was showed that there exists a positive solution for small λ0. When g(u)=u4, Li, Li and Shi [26] proved the existence of positive solutions to (1.2) by using variational method which does not require usual compactness condition. Later, in [27] they studied the existence, nonexistence and multiplicity of positive solutions to (1.2) are influenced on the parameter ranges of λ.

Recently, Azzollini, d'Avenia and Vaira [10] considered the following Schrödinger-Newton type system which is equivalent to a nonlocal version of the well known Brezis Nirenberg problem{Δu=λu+|u|23uϕ,inΩ,Δϕ=|u|21,inΩ,u=ϕ=0onΩ where ΩRN,N3 is a smooth bounded domain. They studied the existence and nonexistence results of positive solutions when N=3 and existence of solutions in both resonance and the non-resonance case for higher dimensions. In [31], Liu studied the following asymptotically periodic Schrödinger-Poisson system with critical exponent{Δu+V(x)uK(x)ϕ|u|3u=f(x,u)inR3,Δϕ=K(x)|u|5,inR3, where V,K,f are asymptotically periodic functions of x. The author proved the existence of positive solutions to (1.4) by the mountain pass theorem and the concentration-compactness principle.

In the special case g(u)=u, system (1.2) reduces to the following well known Schrödinger-Poisson system{Δu+V(x)u+λϕ(x)u=f(x,u)inR3,Δϕ=u2,inR3, which has been studied by many authors, see for example, [7], [4], [6], [18], [16], [25], [34], [36], [37], [38], [43], [44], [47], [48] and the references therein. In [5], [7], [36], the existence and multiplicity of positive solutions were considered for various λ and p; when V depends on x and is not radial, and f is asymptotically linear at infinity, the existence of positive solution for small λ and the nonexistence of nontrivial solution for large λ were obtained in [43]; when V depends on x, the existence of a sign-changing solution was proved in [44]; when V depends on x and is sign-changing, the existence and multiplicity were investigated in [47]; existence of a nontrivial solution and concentration results were showed in [23], [42], [48]. Moreover, the ground state solutions for (1.5) were considered in [9]; the ground and bound state solutions for system (1.5) were studied in [24], [38].

We notice that, in all the papers aforementioned, only few papers like [26], [27], [31] deal with problem (1.4) which is involved with the critical growth for the nonlocal term. The purpose of this paper is to prove that system (1.1) has a ground state solution and relate the number of positive solutions with the topology of the set where V attains its minimum. By using variational method and the Ljusternik-Schnirelmann category theory we shall establish the multiplicity of positive solutions to system (1.1) concentrating at the minimum points set of the potential V, when parameter ε is small enough. To the best of our knowledge, there is not any results for system (1.1) on the existence, multiplicity and concentration of positive solutions in the literature.

We remark that the lack of compactness caused by the unboundedness of the whole space R3 and the critical growth in the nonlocal term ϕu|u|3u (see Section 2), makes the situation more complicated to handle with system (1.1). To overcome these obstacles, we shall transform system (1.1) into a nonlinear Schrödinger equation with a non-local term and apply the variational methods. The compactness involving Palais-Smale sequences are recovered by adopting some more delicate analysis and tricks.

In order to state the main result, we introduce some basics assumptions on the functions V and f. For the potential V, we assume that V:R3R is a continuous function satisfying

  • (V)

    0<V0=infxR3V(x)<liminf|x|V(x):=V.

This kind of hypothesis was first introduced by Rabinowitz [35] in the study of a nonlinear Schrödinger equation, and in this paper we shall consider the case V< or V=. Since we are only concerned with positive solutions of (1.1), we may assume that f:RR is a function of C1 class and satisfies the following conditions:
  • (f1)

    f(s)=0 for all s<0;

  • (f2)

    lims0+f(s)s=0;

  • (f3)

    there exists q(3,5) verifying limsf(s)sq=0;

  • (f4)

    θ>4 such that 0<θF(s):=θ0sf(τ)dτsf(s) for all s>0;

  • (f5)

    the function sf(s)s3 is increasing in (0,).

The assumptions on V and f are quite natural in this context. Assumption (V) was first employed in [35] to take into account potentials which are possibly not coercive. Hypothesis (f1) is not restrictive since we are concerned with positive solutions, and (f2)(f5) are indispensable to use variational techniques which involve in the Palais-Smale condition, the Mountain Pass Theorem and the Nehari manifold. For this aim, we recall that {un} is a Palais-Smale sequence for a C1 functional I at level cR, if I(un)c and I(un)0. We shall abbreviate this by saying that {un} is a (PS)c sequence. Furthermore, the functional I is said to satisfy the Palais-Smale condition at level c, if every (PS)c sequence has a strongly convergent subsequence.

In order to relate the number of solutions of (1.1) with the topology of the set of minima of the potential V, we introduce the set of global minima of V given byM={xR3:V(x)=V0=infxR3V(x)}. In view of (V), the set M is compact. For any δ>0, we denote by Mδ={xR3:dist(x,M)δ} the closed δ-neighborhood of M.

Theorem 1.1

Suppose that f satisfies (f1)(f5) and V verifies (V). Then, for any δ>0, there exists εδ>0 such that, for any ε(0,εδ), problem (1.1) has at least catMδ(M) positive solutions, for any ε(0,εδ). Moreover, if uε denotes one of these positive solutions and ηεR3 its global maximum point, thenlimε0V(ηε)=V0.

We recall that if Y is a closed subset of a topological space X, the Ljusternik-Schnirelmann category catX(Y) (if X=Y we just write cat(X)) is the least number of closed and contractible sets in X which cover Y.

In order to obtain multiple solutions for (1.1), we use some techniques introduced by some papers of Benci, Cerami [11], and Cingolani, Lazzo [17]. The main idea is to make precisely comparisons between the category of some sublevel sets of the energy functional of (1.1) and the category of the set M. For more applications of the Ljusternik-Schnirelmann theory on the study of Schrödinger equations, p-Laplace equations, quasilinear equations, we refer the reader to [1], [2], [20], [21] and references therein.

The paper is organized as follows. In Section 2 we present the abstract framework of the system as well as some preliminary results and present some compactness properties of the functional of the autonomous problem. In Section 3 we prove system (1.1) has a positive ground state solution. Section 4 is devoted to the proof of Theorem 1.1. A technical lemma is given in the Appendix.

As a matter of notation, we denote with Br(y), respectively Br, the ball in RN with radius r>0 centered in y, respectively in 0. The Lp-norm in RN is simply denoted with ||p. If we need to specify the domain, let us say ARN, we write ||Lp(A). From now on, the letter C,C1,i=1,2,, will be repeatedly used to denote various positive constants whose exact values are irrelevant.

Section snippets

Variational framework and notations

Throughout the paper we suppose that the functions V and f satisfy conditions (V) and (f1)(f5), respectively. To fix some notations, we denote the standard norm of H1(R3) by||u||2=R3(|u|2+u2)dx, and the norm of D1,2(R3) by||u||D1,2(R3)2=R3|u|2dx. For every uH1(R3), and any fixed ε>0, the Lax-Milgram theorem implies that there exists a unique ϕuD1,2(R3) such that (e.g. [31])ε2Δϕu=|u|5. Moreover,ϕu(x)=14πε2R3|u(y)|5|xy|dy.

We next summarize some properties about the solution ϕu of the

Existence of a ground state solution

In this section, we show there exists a ground state solution to (2.9), that is, a positive solution uε of (2.9) with Iε(uε)=cε. To study the regularization of the ground state solution, we recall the following two propositions in our case N=3. The first one is an adequate version, for our aim, from a result due to Brezis and Kato [14].

Proposition 3.1

Let uH1(R3) satisfyingΔu+(b(x)q(x))u=f(x,u)inR3, where qL32(R3) and b:R3R+ is a Lloc(R3) function; f:R3×RR+ is a Caratheodory function such that0f(x,s)Cf

Multiplicity of solutions to (2.9)

In this section we are going to prove the multiplicity of solutions and study the behavior of their maximum points in relation to the set M. The main result in this section has the following statement.

Theorem 4.1

Suppose that f satisfies (f1)(f5) and V verifies (V). Then, for any δ>0, there exists εδ>0 such that, for any ε(0,εδ), problem (2.9) has at least catMδ(M) positive solutions, for any ε(0,εδ). Moreover, if uε denotes one of these positive solutions and zεR3 its global maximum point, thenlimε0V

Acknowledgments

The authors are grateful to the referee for careful reading the manuscript and giving valuable comments and suggestions. This work is supported by NSFC (11771468, 11271386).

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