Standing waves for Schrödinger-Poisson system with general nonlinearity

In this paper we consider the following Schrodinger-Poisson system with general nonlinearity \begin{document}$ \begin{eqnarray*} \left\{ \begin{array}{ll} -\varepsilon^2\Delta u+V(x)u+\psi u = f(u),\,\, x\in\mathbb{R}^3,\\ -\varepsilon^2\Delta\psi = u^2,\,\,u>0,\,\, u\in H^1(\mathbb{R}^3),\\ \end{array} \right. \end{eqnarray*} $\end{document} where \begin{document}$ \varepsilon>0 $\end{document} is a small positive parameter. Under a local condition imposed on the potential \begin{document}$ V $\end{document} and general conditions on \begin{document}$ f, $\end{document} we construct a family of positive semiclassical solutions. Moreover, the concentration phenomena around local minimum of \begin{document}$ V $\end{document} and exponential decay of semiclassical solutions are also explored. We do not need the monotonicity of the function \begin{document}$ u\rightarrow\frac{f(u)}{u^3} $\end{document} , and our results include the case \begin{document}$ f(u) = |u|^{p-2}u $\end{document} for \begin{document}$ 3 . Since without more global information on the potential, in the proofs we apply variational methods, penalization techniques and some analytical techniques.

1. Introduction and main results. In this paper, we will consider the following nonlinear Schrödinger-Poisson system −ε 2 ∆u + V (x)u + ψu = f (u), x ∈ R 3 , −ε 2 ∆ψ = u 2 , u > 0, u ∈ H 1 (R 3 ), (1.1) where ε > 0 is a small positive parameter, f is a continuous, superlinear and subcritical nonlinearity. We are interested in the existence and concentration behavior of semiclassical solutions of system (1.1) when ε → 0. The first equation of (1.1) is a nonlinear Schrödinger equation in which the potential ψ satisfies a nonlinear Poisson equation. For this reason, system (1.1) is called Schrödinger-Poisson system. It is well known that the Schrödinger-Poisson system (1.1) has some strong physical meaning. For instance, in Abelian Gauge Theories, system (1.1) provides a model to describe the interaction of a nonlinear Schrödinger field with the electromagnetic field. In quantum electrodynamics, system (1.1) describes the interaction between a charge particle interacting with the electromagnetic field. Moreover, system (1.1) also appears in semiconductor theory, nonlinear optics and plasma physics (see [4,23,25]). For more details in the physical backgrounds, we refer the readers to [3,4].
For ε > 0 small, problem (1.1) is called perturbed problem. In this case the solutions of (1.1) are referred to as semiclassical solutions, which describes the transition from quantum mechanics to classical mechanics when the parameter ε goes to zero, and possess an important physical interest. In this framework, from a mathematical viewpoint, one is interested not only in existence of semiclassical solutions but also in their asymptotic behavior as ε → 0. Typically, solutions tend to concentrate around critical points of the potentials functions: such solutions are called spikes. In the present paper we only care about the perturbed problem, so in the following we shall recall some previous results for this case. In [18], Ianni and Vaira considered the following system (

1.2)
Under some subcritical conditions on f , the authors proved that (1.2) has a singlebump solution near critical points of the potential A. Later, by applying a standard Lyapunov-Schmidt reduction methods, Ruiz and Vaira [32] studied the existence of multi-bump solutions, whose bumps concentrated around a local minimum of the potential A. For other related results, we refer the readers to [12,19,20,31] for concentration on spheres.
Observe that, the singular perturbation problem (1.1) has two perturbation parameters in the system, and is called the double parameters' perturbation problem. Here we would like to bring attention to the readers that there is a big difference between system (1.1) and (1.2). Indeed, when the parameter ε is small, the limit problem for the latter is the classical nonlinear Schrödinger equation while the former is still the Schrödinger-Poisson system. Recently, problem (1.1) was also considered in [15,16,17,39,40,43]. More specifically, under the subcritical and Ambrosetti-Rabinowitz type growth condition, He [16] studied the existence of positive solutions by using minimax theorems, and proved that these solutions concentrate around the global minimum of the potential V in the semiclassical limit. Moreover, a multiplicity result depending on the topology property of the potential is obtained by the Ljusternik-Schnirelmann theory. Here the author assumed that and f is a C 1 function such that f (s)s 2 − 3f (s)s ≥ cs σ , σ ∈ (4, 6), c > 0 and f (s) = o(s 3 ) as s → 0.
(1.4) Subsequently, this result has been extended to the critical growth case by He and Zou [16]. Also under the global condition on the potential functions, Wang et al. [39] considered the following system with competing potentials The authors proved the existence of semi-classical ground state solutions by the Nehari manifold method under the following conditions (1.5) And moreover, some new concentration phenomenons of semi-classical solutions on the minimum points of a(x) and the maximum points of b(x) are also investigated. For the critical case, we refer readers to [40] for details. It is worth pointing out that the global condition (1.3) used in [15,16,39,40] plays a crucial role in proving the existence of positive solutions. Indeed, the key point is that the property of the potential V can help us to restore the necessary compactness by comparing energy levels of original problem and limit problem. An interesting question is whether one can find solutions which concentrating around local minima of the potential. Very recently, it seems that the only work concerning the existence and concentration behavior of solutions is due to He and Li [17] for the local potential condition. In [17], the authors considered the critical growth model f (u) = λ|u| p−2 u + |u| 4 u, λ > 0 and 3 < p ≤ 4, and assumed that the following local condition first introduced by del Pino and Felmer in [27]: Using a version of quantitative deformation lemma due to Figueiredo, Ikoma and Santos Junior [13], they constructed a special bounded Palais-Smale sequence and recovered the compactness by using a penalization method which was introduced in [6].
Motivated by the above facts, in the present paper we focus on the existence and concentration behavior of semiclassical solutions of system (1.1) under the local potential and general subcritical nonlinearity. To the best of our knowledge, it seems that such a problem was not considered in literature before. More precisely, our purpose in this paper is twofold. First, we construct a family of positive semiclassical solutions for (1.1) with some properties, such as concentration, exponent decay, etc. Second, we treat more general subcritical nonlinearity f that is weaker than the previous works [15,17,39].
In order to describe some concentration phenomena of semiclassical solution, we define the concentration set by M := {x ∈ Λ : V (x) = V 0 }. Moreover, without loss of generality, we may assume that 0 ∈ M. We are now in a position to state the main result of this paper.
Moreover, u ε (εx + x ε ) converges (up to a subsequence) locally uniformly to a positive ground state solution of limiting system where C 1 , C 2 are independent of ε.
The proof will be given in Lemma 3.1. These relation will be used later in proving the existence of positive ground state solution of limiting problem. Remark 1.3. Compared with [15] and [39], on the one hand, our condition (V 2 ) is rather weak, without restriction on the global behavior of V is required other than (1.3), and the behavior of V outside Λ is irrelevant in this paper. On the other hand, the conditions we assumed for the nonlinearity f can not satisfy the classical Ambrosetti-Rabinowitz condition in (1.4) and the 4-superlinear condition in (1.5) and the monotonicity condition of the function f (s)/s 3 . Instead, f only satisfies the superlinear condition at origin and the general 3-superlinear condition at infinity, and a weaker monotonicity condition in this paper. Clearly, ours conditions are weaker than the ones in [15,39]. Besides, in [16] the nonlinearity f is only a special power nonlinear model with critical term. But our conditions seem more general and are subcritical growth. In particular, our results include the case f (u) = |u| p−2 u for 3 < p < 6. From these reasons, our results extend and complement related results [15,16,39].
The motivation of the present paper mainly comes from the results of semiclassical Schrödinger equations (1.6) Many mathematicians studied the existence, concentration and multiplicity of solutions for (1.6). In [29], Rabinowitz used the Mountain-Pass theorem to show that (1.6) possesses a positive ground state solution for ε > 0 small under the global condition (1.3). After that, the authors in [38] showed that the positive solution obtained in [29] concentrates at global minimum points of V . Under the local conditions (V 1 )-(V 2 ), del Pino and Felmer [27] introduced a penalization approach, so-called local mountain pass, and managed to handle the case of a, possibly degenerate, local minimum of V . They showed the existence of a single spike solution concentrating around minimizer of V in Λ. Based on a new singular perturbation, essentially a penalization method, the localized bound state solutions concentrating at an isolated component of the local minimum of V under the almost optimal conditions on f (such as Berestycki-Lions condition) were also constructed in papers [5] and [6]. From the commentaries above, it is quite natural to ask that: can we obtain some similar results for the Schrödinger-Poisson system (1.1) as in Schrödinger equation (1.6)? In the present paper, we shall give some answers for this system. However, compared with the Schrödinger equation (1.6), the Schrödinger-Poisson system (1.1) becomes more complicated since there exists a competing effect of the non-local term with the nonlinear term. Hence our problem poses more challenges in the calculus of variation. In addition, there are also some additional difficulties for system (1.1). (i) The nonlinearity f does not satisfy Ambrosetti-Rabinowitz condition and the fact that the function f (s)/s 3 is not increasing on (0, ∞), these features prevent us from obtaining a bounded Palais-Smale sequence ((PS) sequence in short) and using the usual Nehari manifold method. (ii) The unboundedness of the domain R 3 leads to the lack of compactness. As we shall see later, the above two aspects prevent us from using variational method in a standard way.
Our argument is based on variational method, which can be outlined as follows. The solutions are obtained as critical points of the energy functional associated to system (1.1). Moreover, the above mentioned difficulties we should overcome. Since the monotonicity of the function f (s)/s 3 is not required, which gives rise to the Nehari manifold method can not be applied in this paper. On the one hand, we need to use a penalization method introduced by Byeon and Wang [6], which helps us to overcome the obstacle caused by the non-compactness due to the unboundedness of the domain and the lack of Ambrosetti-Rabinowitz condition. Proceeding by some arguments and techniques, then the existence of ground state solution follows. On the other hand, inspired by [13,17], use a version of quantitative deformation lemma due to Figueiredo, Ikoma and Junior [13] to construct a special bounded (PS) sequence in a neighborhood of the compact set.
More precisely, for the proof of our results, some arguments are in order. Firstly, in order to get concentrated solutions to system (1.1) we need to consider the existence of ground state solutions of the associated "limiting problem" of system (1.1): with the corresponding energy functional Denoting S a the set of ground state solution U of (1.7) satisfying U (0) = max x∈R 3 U (x), we will show that S a is compact in H 1 (R 3 ). To conquer the difficulties of certifying the boundedness of (PS) sequence for the limiting equation, we consider an auxiliary functional G a (u) = 2 I a (u), u − P a (u), where P a (u) = 0 is the Pohozaev identity of (1.7). By applying the ideas employed by Jeanjean [21] and Tang and Chen [35,36], we find a (PS) sequence where c a > 0 is mountain pass level defined later by (3.10).
To investigate (1.1), we will focus on the following equivalent system by making the change of variable εy = x with the energy functional Different to [15] and [39], where the potential V has the global condition (1.3) and the nonlinear term f satisfies the Ambrosetti-Rabinowitz condition or the general 4-superlinear condition, in this case the mountain geometry of the functional and boundedness and compactness of (PS) sequences can be obtained easily. Here in the present paper, all conditions we assumed are weaker than the ones in [15] and [39], namely, the potential condition (V 2 ) is local and the nonlinear term f satisfies the general 3-superlinear condition and a weaker monotonicity condition. Then some arguments used in [15] and [39] are no longer valid, and some new methods and techniques need to be introduced in this paper. All we do is to build a modification of the energy functional associated to (1.1). In such a way, the functional is proved to satisfy the so-called (PS) condition. The modification of the energy functional corresponds to a penalization technique "outside Λ", that is why no other global condition are required for the potential V . Following [6], we set J ε : H ε → R be given by It will be shown that the functional Q ε acts as a penalization to force the concentration phenomena to occur inside Λ. It is standard to see that J ε ∈ C 1 (H ε , R). Clearly a critical point of I ε corresponds to a solution of system (1.8). To find solutions of system (1.8) which concentrate in Λ as ε → 0, we shall search critical points of the modified functional J ε for which Q ε is zero. To do this, we need to establish a delicate L ∞ -estimation for the critical point v ε of J ε , and moreover show the critical point v ε of J ε is indeed a solution to the original problem.
The paper is organized as follows. In Section 2, we give some preliminary lemmas. In Section 3, we analyze the "limiting problem" and show the existence of positive ground state solution and compactness of S a . In section 4, we prove the main result Theorem 1.1.
Throughout the sequel, we denote the norms of usual Lebesgue space L p (R 3 ) by and C i and C denotes different positive constant in different places.

2.
Preliminaries. In this section, we recall that by the Lax-Milgram theorem, for Moreover, ψ u can be expressed as The function ψ u has the following properties, the proof can be found in [7] and [30].
(iv) If y ∈ R 3 andũ = u(x + y), then ψũ(x) = ψ u (x + y) and is weakly sequentially continuous; 3. The limiting problem. We will make use of the limiting problem for proving our main result. To this end, we first discuss in this section the existence of the positive ground state solutions of the limiting problem. Consider the limiting system of (1.1) as follows where a > 0. Since we need to seek a positive ground state solution, we restrict the nonlinearity f (s) = 0 if s ≤ 0. In view of [28], if u ∈ H 1 (R 3 ) is a weak solution to problem (1.7), then we have the following Pohozaev's identity: As in [30], we introduce the following Nehari-Phozaev manifold Proof. It is obvious that (3.2) holds for τ ≤ 0. For τ > 0, let Then from (f 4 ) and Remark 1.2, one has (3.4) It follows that g(t) ≥ g(1) = 0 for t > 0. Then we complete the proof. Moreover, let t → 0 in (3.2), so we complete the proof of (f 5 ) in Remark 1.2.
From Lemma 3.2, we get the following corollary immediately.
Proof. Suppose by contradiction that the lemma does not hold, then by Lion's concentration compactness principle [24], for 2 < p < 6, Using (f 1 ), (f 2 ) and I a (u n ), u n = o(1), we get By (3.11), we have I a (u n ) → c a = 0, which contradicts to c a > 0. Therefore we complete the proof.
Thus we complete the proof. Proof. Let {u n } be the sequence given in (3.10) and c a be the Mountain-Pass value for I a . Setũ n (x) = u n (x + ξ n ), where {ξ n } is the sequence given by in Lemma 3.8. Up to a subsequence, we havẽ u n ũ weakly in H 1 (R 3 ), u n →ũ strongly in L p loc (R 3 ) for 1 ≤ p < 6, u n →ũ a.e. in R 3 , and G a (ũ) = 0. By (f 5 ), Lemma 3.9 and Fatou's Lemma, one has Hence, I a (ũ) = c a and I a (ũ) = 0. Moreover, by the standard elliptic estimate and strong maximum principle (see [14]),ũ > 0 for all x ∈ R 3 , soũ is a positive ground state solution of (1.7). Thus, we complete the proof.
Let S a the set of ground state solutions U of (1.7) satisfying U (0) = max x∈R 3 U (x). Then, we get the following compactness of S a .
Proof. For any U ∈ S a , similar to the proof of Lemma 3.7, we can get S a is bounded in H 1 (R 3 ). For any sequence {U k } ⊂ S a , up to a subsequence, we may assume that there is and U 0 satisfies Next, we will show that U 0 is nontrivial. Using Brezis-Kato's type arguments, we can get U k L p loc (R 3 ) ≤ C for 1 < p < ∞. By the classical Calderón-Zegmund L p regularity estimates (see Theorem 9.9, [14]) and Morry-Sobolev embedding theorem, we can get that U k C 1,α loc (R 3 ) ≤ C for some α ∈ (0, 1). Then using Schauder's estimate, we have Since ∆U k (0) ≤ 0, from (1.7), we can check that there exists b > 0 such that which means that I a (U 0 ) = c a and (3.14) Similar to the proof of Lemma 3.7, we can deduced { U k H 1 (R 3 ) } is bounded. Using the interpolation inequality, we get On the other hand, We can deduce that Using Lemma 2.1, Lemma 2.2 and (3.15), it is standard to prove that Next, we claim that lim sup |x|→∞ U k (x) = 0 uniformly for all k. To the contrary, we assume that there exist δ > 0 and Combining with (3.16), we obtain So we prove the claim. Then by the comparison principle and elliptic estimates, we conclude exist two positive constants C 3 and C 4 such that This completes the proof that S a is compact in H 1 (R 3 ).

4.
Proof of the Theorem 1.1. For the proof of our results, we do not handle system (1.1) directly, but instead we handle an equivalent system to (1.1). For this purpose, making the change of variable εy = x, then (1.1) can be rewritten as It is standard to show that J ε ∈ C 1 (H ε , R). To find solutions of (1.8) which concentrate around the local minimum of V in Λ as ε → 0, we shall search critical points of J ε for which Q ε is zero. Let c V0 = I V0 (w) for w ∈ S V0 and 10δ = dist{M, R 3 \ Λ}, we fix a β ∈ (0, δ) and a cut-off function ϕ ∈ C ∞ c (R 3 ) such that 0 ≤ ϕ ≤ 1, ϕ(x) = 1 for |x| ≤ β, ϕ(x) = 0 for |x| ≥ 2β and |∇ϕ| ≤ C β . We will find a solution of (1.8) near the set for sufficiently small ε > 0, where we use the notation M β := {y ∈ R 3 : inf z∈M |y − z| ≤ β}. For A ⊂ H ε , we define A a := {u ∈ H ε : inf v∈A u − v Hε ≤ a}.
The next lemma plays a key role in proving the Theorem 1.1.
there exists, up to a subsequence, (ii) If we drop {R εi } and replace (4.4) by then the same conclusion holds.
Since u εj (x) = w εj ( x εj − y j ), x j := ε j P j + ε j y j is a maximum point of u εj . Using (4.38), x j → x 0 ∈ M as j → ∞. Since the sequence {ε j } is arbitrary, we get the existence and concentration results in Theorem 1.1. The proof of the exponential decay of u ε is standard (see [15] and [22]), we omit it here. Thus we complete the proof of Theorem 1.1.