Existence of positive multi-bump solutions for a Schr\"odinger-Poisson system in $\mathbb{R}^{3}$

In this paper we are going to study a class of Schr\"odinger-Poisson system $$ \left\{ \begin{array}{ll} - \Delta u + (\lambda a(x)+1)u+ \phi u = f(u) \mbox{ in } \,\,\, \mathbb{R}^{3},\\ -\Delta \phi=u^2 \mbox{ in } \,\,\, \mathbb{R}^{3}.\\ \end{array} \right. $$ Assuming that the nonnegative function $a(x)$ has a potential well $int (a^{-1}(\{0\}))$ consisting of $k$ disjoint components $\Omega_1, \Omega_2, ....., \Omega_k$ and the nonlinearity $f(t)$ has a subcritical growth, we are able to establish the existence of positive multi-bump solutions by variational methods.


Introduction
This paper was motivated by some recent works concerning the nonlinear Schrödinger-Poisson system where V : R 3 → R is a nonnegative continuous function with inf x∈R 3 V (x) > 0, 2 < p < 2 * = 6 and ψ : Ω → C and φ : Ω → R are two unknown functions. The first equation in system (N SP ), called Schrödinger equation, describes quantum (non-relativistic) particles interacting with the eletromagnetic field generated by the motion. An interesting phenomenon of this Schrödinger type equation is that the potential φ(x) is determined by the charge of wave function itself, that means, φ(x) satisfies the second equation (Poisson equation) in system (N SP ).
As we all know, the knowledge of the solutions for the elliptic system has a great importance in studying stationary solutions ψ(x, t) = e −it u(x) of (N SP ). It is convenient to observe that the system (SP ) contains two kinds of nonlinearities. The first one is φ(x)u which is nonlocal, since the electrostatic potential φ(x) depends also on the wave function, and is used to describe the interaction between the solitary wave and the electric field. The second type of nonlinearity f (u) is a local one which has been used to model the external forces involving only functions of fields. For more information about the physical background of system (SP ), we cite the papers of Benci-Fortunato [6], Bokanowski & Mauser [8], Mauser [24], Ruiz [26], Ambrosetti-Ruiz [4] and S'anchez & Soler [28]. An important fact for system (SP ) is that it can be reduced into one single Schrödinger equation with a nonlocal term (see, for instance, [5,17,26,30]). Effectively, by the Lax-Milgram Theorem, given u ∈ H 1 (R 3 ), there exists an unique φ = φ u ∈ D 1,2 (R 3 ) such that −∆φ = u 2 in R 3 .
By using standard arguments, we know that φ u verifies the following properties (for a proof see [11,26,30]): ii) there exists C > 0 Therefore, (u, φ) ∈ H 1 (R 3 ) × D 1,2 (R 3 ) is a solution of (SP ) if, and only if, u ∈ H 1 (R 3 ) is a solution of the nonlocal problem where φ u = φ ∈ D 1,2 (R 3 ). Now, we would like to emphasize that the existence of solutions for problem (P ) can be established via variational methods. Associated to the elliptic equation (P ), we have the energy functional I : E → R given by where F (s) = s 0 f (t)dt and E is the function space Supposing some conditions on f , Lemma 1.1 implies that the functional I is well defined and I ∈ C 1 (E, R) with Hence, the critical points of functional I are in fact the weak solutions for nonlocal problem (P ).
From the above commentaries, we know that the system (SP ) has a nontrivial solution if, and only if, the nonlocal problem (P ) has a nontrivial solution. In the last years, many authors had studied the system (SP ) and focused their attentions to establish existence and nonexistence of solutions, multiplicity of solutions, ground state solutions, radial and nonradial solutions, semiclassical limit and concentrations of solutions, see for example the papers of Azzollini & Pomponio [5], Cerami & Vaira [9], Coclite [10], D'Aprile & Mugnai [11,12], d'Avenia [13], Ianni [19], Kikuchi [18], and Zhao & Zhao [30]. For the problem set in a bounded domain, we would like to cite the papers of Siciliano [17], Ruiz & Siciliano [27] and Pisani & Siciliano [25] for nonnegative solutions and Alves & Souto [3], Ianni [20] and Kim & Seok [22] for sign-changing solutions. However, related to the existence of multi-bump solutions for Schrödinger-Poisson system with potential well, as far as we know, there seems to be no existing results.
In the present paper, we will assume that potential V (x) is of the form where λ is a positive parameter and a : R 3 → R is a nonnegative continuous function. Hence, the problems (SP ) and (P ) can be written respectively as To state the main result, we assume that the function a(x) verifying the following conditions: Related to the function f , we will assume the ensuing conditions: Moreover, we also assume that the nonlinearity f satisfies The motivation to investigate problem (SP ) λ goes back to the papers [1] and [15]. In [15], inspired by [14] and [29], the authors considered the existence of positive multi-bump solution for the problem The authors showed that the above problem has at least 2 k − 1 solutions u λ for large values of λ. More precisely, for each non-empty subset Υ of {1, . . . , k}, it was proved that, for any sequence λ n → ∞ we can extract a subsequence (λ n i ) such that (u λn i ) converges strongly in H 1 R N to a function u, which satisfies u = 0 outside Ω Υ = j∈Υ Ω j and u | Ω j , j ∈ Υ, is a least energy solution for (1.5) After, in [1], Alves extended the results described above to the quasilinear Schrödinger equation driven by p-Laplacian operator.
Involving the Schrödinger-Poisson system with potential wells, there are not so many existing papers. As far as we know, the only paper that considered the existence of solutions for system (SP ) λ is due to Jiang and Zhou [21] where the authors studied the existence and properties of the solutions depending on some parameters. However, nothing is known for the existence of multi-bump type solutions. Motivated by the above references, we intend in the present paper to study the existence of positive multi-bump solution for (SP ) λ . However, we need to point out some difficulties involving this subject: 1-It is well known that the equation (1.5) plays the role of limit equation for (1.4) as λ goes to infinity and the ground state solution of (1.5) plays an important role in building the multi-bump solutions for (1.4). However, little is known about what is the corresponding limit equation for equation (P ) λ when the parameter λ goes to infinity. 2-Once discovered the limit problem for equation (P ) λ , it is crucial to prove that it has a specially shaped least energy solution on a subset of the Nehari manifold , see Section 2 for more details.
3-When we apply variational methods to prove the existence of solution to (SP ) λ , we are led to study a nonlocal, see problem (P ) λ above. However, for this class of problem, it is necessary to make a careful revision in the sets used in the deformation lemma found in [1] and [15] to get multi-bump solution, since they don't work well for this class of system, see Sections 6 and 7 for more details.
(P ) ∞,Υ In the proof of Theorem 1.2, we need to study the existence of least energy solution for problem (P ) ∞,Υ . The main idea is to prove that the energy function J associated with nonlocal problem (P ) ∞,Υ given by assumes a minimum value on the set where u j = u | Ω j and N Υ is the corresponding Nehari manifold defined by More precisely, we will prove that there is w ∈ M Υ such that After, we use a deformation lemma to prove that w is a critical point of J, and so, w is a least energy solution for (P ) ∞,Υ . The main feature of the least energy solution w is that w(x) > 0 ∀x ∈ Ω j and ∀j ∈ Υ which will be used to describe the existence of multi-bump solutions.
Since we intend to look for positive solutions, through this paper we assume that f (s) = 0, s ≤ 0.

The problem (P ) ∞,Υ
In what follows, to show in details the idea of proving the existence of least energy solution for (P ) ∞,Υ , we will consider Υ = {1, 2}. Moreover, we will denote by Ω, N and M the sets Ω Υ , N Υ and M Υ respectively. Thereby, with u j = u | Ω j , j = 1, 2. Since we want to look for least energy for (P ) ∞,Υ , our goal is to prove the existence of a critical point for J in the set M.
In order to show that the set M is not empty, we need of the following Lemma.
Proof. It what follows, we consider the vector field .
, a straightforward computation yields that there are 0 < r < R such that and Now, the lemma follows by applying Miranda theorem [23].
As an immediate consequence of the last lemma, we have the following corollary Corollary 2.2. The set M is not empty.
Next, we will show some technical lemmas.
Proof. From (f 1 ) and (f 2 ), given ε > 0 there exists C > 0 such that Since w n ∈ M, by Lemma 2.3 that is, Using the boundedness of (w n ), there is C 1 such that showing that

Existence of least energy solution for (P ) ∞,Υ
In this subsection, our main goal is to prove the following result Proof. In what follows, we denote by c 0 the infimum of J on M, that is, From Lemma 2.3(i), we conclude that c 0 > 0. By Corollary 2.2, we know that M is not empty, then there is a sequence (w n ) ⊂ M satisfying lim n J(w n ) = c 0 .
Still from Lemma 2.3(i), (w n ) is a bounded sequence. Hence, without loss of generality, we may suppose that there is w ∈ H 1 0 (Ω) verifying Then, (f 2 ) combined with the compactness lemma of Strauss [7, Theorem A.I, p.338] gives from where it follows together with Lemma 2.4 that w j = 0 for j = 1, 2. Then, by Lemma 2.1 there are t, s > 0 verifying Next, we will show that t, s ≤ 1. Since J ′ (w n,j )w n,j = 0 for j = 1, 2, Taking the limit in the above equalities, we obtain Recalling that it follows that Now, without loss of generality, we will suppose that s ≥ t. Under this condition, If s > 1, the left side in this inequality is negative, but from (f 4 ), the right side is positive, thus we must have s ≤ 1, which also implies that t ≤ 1. Our next step is to show that J(tw 1 + sw 2 ) = c 0 . Recalling that tw 1 + sw 2 ∈ M, we derive that Hence, . and leading to Using Fatous' Lemma together with (f 4 ), we see that Until this moment, we have proved that there exists a To complete the proof of Theorem 1.2, we claim that w is a critical point for functional J. To see why, for each ϕ ∈ H 1 0 (Ω), we introduce the functions Q i : R 3 → R, i = 1, 2 given by By a direct computation, and so, By (f 4 ), we know that f ′ (s)s 2 ≥ 3f (s)s for all s ≥ 0. Thus, Now, recalling that J ′ (w)w 1 = 0, we have Then, The same type of argument gives Therefore, applying the implicit function theorem, there are functions z(r), l(r) of class C 1 defined on some interval (−δ, δ), δ > 0 such that z(0) = l(0) = 0 and This shows that for any t ∈ (−δ, δ), Since, we derive that J(v(r)) ≥ J(w), ∀r ∈ (−δ, δ), that is, J(w + rϕ + z(r)w 1 + l(r)w 2 ) ≥ J(w), ∀r ∈ (−δ, δ).
From this, Taking the limit of r → 0, we get Recalling that J ′ (w)w 1 = J ′ (w)w 2 = 0, the above inequality loads to and so, J ′ (w)ϕ = 0, ∀ϕ ∈ H 1 0 (Ω), showing that w is a critical point for J.

An auxiliary problem
In this section, we work with an auxiliary problem adapting the ideas explored by del Pino & Felmer in [14] (see also [1] and [15]).
We start recalling that the energy functional I λ : E λ → R associated with (P ) λ is given by . .
We recall that for any ǫ > 0, the hypotheses (f 1 ) and (f 2 ) yield Consequently, where C ǫ depends on ǫ. Moreover, for ν > 0 fixed in (3.2), the assumptions (f 1 ) and (f 4 ) imply that there is an unique a > 0 verifying Using the numbers a and ν, we set the functionf : R 3 × R → R given bỹ which fulfills the inequalityf (s) ≤ ν|s|, ∀s ∈ R.
is formed by k connected components Ω 1 , . . . , Ω k with dist Ω i , Ω j > 0, i = j, then for each j ∈ {1, . . . , k}, we are able to fix a smooth bounded domain Ω ′ j such that From now on, we fix a non-empty subset Υ ⊂ {1, . . . , k} and Using the above notations, we set the functions and the auxiliary nonlocal problem The problem A λ is related to P λ in the sense that, if u λ is a solution for A λ verifying then it is a solution for P λ . In comparison to P λ , problem A λ has the advantage that the energy functional associated with A λ , namely, φ λ : E λ → R given by satisfies the (P S) condition, whereas I λ does not necessarily satisfy this condition. Proof. Let (u n ) be a (P S) d sequence for φ λ . So, there is n 0 ∈ N such that On the other hand, by (3.7) and (3.8) which together with (3.2) gives Hence, once that g has a subcritical growth, if u ∈ E λ is the weak limit of (u n ), then where C > 0 does not depend on R. This way, it follows from (3.7), Using Hölder's inequality, we derive Since (u n ) and (|∇u n |) are bounded in L 2 (R 3 ), we obtain So, given ǫ > 0, choosing a R > 0 possibly still bigger, we have that C (1 − ν)R < ǫ, which proves (3.10). Now, we will show that Using the fact that g(x, u)u ∈ L 1 (R 3 ) together with (3.10) and Sobolev embeddings, given ǫ > 0, we can choose R > 0 such that On the other hand, since g has a subcritical growth, we have by compact embeddings Combining the above information, we conclude that The same type of arguments works to prove that Proof Let (u n ) be a (P S) d sequence for φ λ and u ∈ E λ such that u n ⇀ u in E λ . Thereby, by Proposition 3.2, Moreover, the weak limit also gives Recalling that φ ′ λ (u n )u n = o n (1) and φ ′ λ (u n )u = o n (1), the above limits lead to u n − u 2 λ → 0, finishing the proof.

The (P S) ∞ condition
for some c ∈ R.
Proof. Using the Proposition 3.1, we know that ( u n λn ) is bounded in R and (u n ) is bounded in H 1 (R 3 ). So, up to a subsequence, there exists u ∈ H 1 (R 3 ) such that u n ⇀ u in H 1 (R 3 ) and u n (x) → u(x) for a.e. x ∈ R 3 . Now, for each m ∈ N, we define C m = x ∈ R 3 ; a(x) ≥ 1 m . Without loss of generality, we can assume λ n < 2(λ n − 1), ∀n ∈ N. Thus By Fatou's lemma, we derive Cm |u| 2 dx = 0, which implies that u = 0 in C m and, consequently, u = 0 in R 3 \ Ω. From this, we are able to prove (i) − (vi).
(vi) We can write the functional φ λn in the following way

The boundedness of the A λ solutions
In this section, we study the boundedness outside Ω ′ Υ for some solutions of A λ . To this end, we adapt the arguments found in [1] and [16] for our new setting.
Proof. Since ∂Ω ′ Υ is a compact set, fixed a neighborhood B of ∂Ω ′ Υ such that the interation Moser technique implies that there is C > 0, which is independent of λ, such that Hence, there is λ * > 0 such that and so, In fact, extending u λ = 0 in Ω ′ Υ and taking u λ as a test function, we obtain Now, by (3.6), This form, u λ = 0 in R 3 \ Ω ′ Υ + . Obviously, u λ = 0 at the points where u λ ≤ a, consequently, u λ = 0 in R N \ Ω ′ Υ .
By a direct computation, it is possible to show that there is τ > 0 such that if u ∈ M Υ , then u j j > τ, ∀j ∈ Υ, (6.1) where, j denotes the norm on H 1 0 (Ω j ) given by In particular, since w Υ ∈ M Υ , we also have where w Υ,j = w Υ | Ω j for all j ∈ Υ.
By the previous item, c λn,Υ is bounded. Then, there exists (w n k ) subsequence of (w n ) such that (φ λn k ,Υ (w n k )) converges and φ ′ λn k ,Υ (w n k ) = 0. Now, repeating similar arguments explored in the proof of Proposition 4.1, there is w ∈ , as k → ∞. Furthermore, we also can prove that c λn k ,Υ = φ λn k ,Υ (w n k ) → I Υ (w) and Then, w ∈ M Υ , and by definition of c Υ , The last inequality together with item (i) implies This establishes the asserted result.

Proof of the main theorem
To prove Theorem 1.2, we need to find nonnegative solutions u λ for large values of λ, which converges to a least energy solution of (P ) ∞,Υ as λ → ∞. To this end, we will show two propositions which together with the Propositions 4.1 and 5.1 will imply that Theorem 1.2 holds. Henceforth, we denote by Moreover, we fix δ = τ 48R , and for µ > 0, We observe that Proof We assume that there exist λ n → ∞ and u n ∈ A λn Since u n ∈ A λn 2µ , this implies ( u n λn ) is a bounded sequence and, consequently, it follows that φ λn (u n ) is also bounded. Thus, passing a subsequence if necessary, we can assume (φ λn (u n )) converges. Thus, from Proposition 4.1, there exists 0 ≤ u ∈ H 1 0 (Ω Υ ) such that u is a solution for (SP ) Υ , Recalling that (u n ) ⊂ Θ 2δ , we derive that ∀j ∈ Υ.
The item (b) ensures we can use Proposition 5.1 to deduce u λn is a solution for (SP ) λn , for large values of n, which is a contradiction, showing this way the claim. Now, our goal is to prove the second part of the theorem. To this end, let (u λn ) be a sequence verifying the above limits. A direct computation gives φ λn (u λn ) → d with d ≤ c Υ . This way, using Proposition 4.1 combined with item (c), we derive (u λn ) converges in H 1 (R 3 ) to a function u ∈ H 1 (R 3 ), which satisfies u = 0 outside Ω Υ and u | Ω j = 0, j ∈ Υ, and u is a positive solution for      −∆u + u + Ω Υ u 2 |x − y| dy u = f (u), in Ω Υ , u ∈ H 1 0 (Ω Υ ), (P ) ∞,Υ and so, On the other hand, we also know that φ λn (u λn ) → I Υ (u), implying that I Υ (u) = d and d ≥ c Υ .