Non-trivial solutions for Schrödinger-Poisson systems involving critical nonlocal term and potential vanishing at infinity

Abstract The present study is concerned with the following Schrödinger-Poisson system involving critical nonlocal term −△u+V(x)u−l(x)ϕ|u|3u=ηK(x)f(u),  in  R3,−△ϕ=l(x)|u|5,                                   in  R3, $$\begin{array}{} \displaystyle \begin{cases} -\triangle u+V(x)u-l(x)\phi|u|^{3}u=\eta K(x)f(u),~~\mbox{in}~~\mathbb{R}^{3}, \notag\\ -\triangle\phi=l(x)|u|^{5},~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\mbox{in}~~\mathbb{R}^{3}, \end{cases} \end{array}$$(1.1) where the potential V(x) and K(x) are positive continuous functions that vanish at infinity, and l(x) is bounded, nonnegative continuous function. Under some simple assumptions on V, K, l and f, we prove that the problem (1.1) has a non-trivial solution.


Introduction and main results
The aim of this paper is to investigate the existence of non-trivial solutions for the following Schrödinger-Poisson system involving critical nonlocal term and potential vanishing at in nity where V , K ∈ C(R , R), f ∈ C(R × R, R), l(x) is bounded, nonnegative continuous function, and V , K are nonnegative functions which can be vanishing at in nity. η > is a parameter and * := is the critical Sobolev exponent. Similar problems have been widely investigated, and it is well known that they have a strong physical meaning, because they appear in quantum mechanics models and in semiconductor theory. In particular, systems like ( . ) have been introduced in [1] as a model describing solitary waves, for nonlinear stationary equations of Schrödinger type interacting with an electrostatic eld, and are usually known as Schrödinger-Poisson systems. Indeed, in ( . ) the rst equation is a nonlinear stationary Schrödinger equation that is coupled with a Poisson equation, to be satis ed by ϕ, meaning that the potential is determined by the charge of the wave function. For more details, we refer the readers to [2][3][4][5] and the references therein.
In [22], A. Azzollini and P. d'Avenia rstly studied the following Schrödinger-Poisson system with critical nonlocal term They proved that the existence and nonexistence results for system ( . ) depend on the value of λ.
In [23], Liu studied the following Schrödinger-Poisson system with critical nonlocal term Under the condition V(x), K(x), f are asymptotically periodic, the author proved the system (1.4) has at least a positive solution by the mountain pass theorem and the concentration-compactness principle.
In [25], F. Li, Y. Li and J. Shi proved possesses at least one positive radially symmetric solution when b > is a constant. To the best of our knowledge, there seems to be little progress on the existence of nontrivial solution for Schrödinger-Poisson systems involving critical nonlocal term and potential vanishing at in nity. By the motivation of above work, In our article, we establish the existence of non-trivial solution for problem ( . ) with critical nonlocal term and potential vanishing at in nity. Firstly the critical growth causes a lack of compactness, and it is much more di cult to obtain the existence of non-trivial solutions. Secondly since V(x) is potential vanishing at in nity, which makes our studies more interesting. At last, we obtain a non-trivial solution by using the mountain pass theorem without (PS) condition.
Below, we assume that the pair (V , K) of continuous functions V , K : R → R belongs to K. Throughout the paper, (V , K) ∈ K means that (VK ): V(x), K(x) > for all x ∈ R and K ∈ L ∞ (R ). Furthermore, one of the below conditions occurs: The hypotheses (VK ) − (VK ) on functions V and K were rst introduced in [26] and characterized problem ( . ) as zero mass. Problems of zero mass have been studied by many authors, see for example, [27][28][29][30][31][32] and references therein. Finally, we assume the following growth conditions at the origin and at the in nity for the continuous function f : R → R:  (f ) There exists a θ ∈ ( , * ) such that Furthermore, we make the following hypotheses on the function l(x). (l ) There exists x , such that l(x ) = sup x∈R l(x).
(l ) For x close to x we have Now we state our main results as follows.
, for su ciently lange η > , then system ( . ) has at least one non-trivial solution.
Notation. In this paper we make use of the following notations: C will denote various positive constants; the strong (respectively weak) convergence is denoted by → (respectively ); o( ) denotes o( ) → as n → ∞, Bρ( ) denotes a ball centered at the origin with radius ρ > . The remainder of this paper is organized as follows. In Section 2, some preliminary results are presented. In Section 3, we give the proof of our main results.

Variational setting and preliminaries
Let us consider the space Recall that a weak solution of problem ( . ) is a function u ∈ E such that for all φ ∈ E. Then, the weak solutions of ( . ) are the critical points of the energy functional de ned on E by where F(u) = u f (s)ds. More precisely, J ∈ C (E, R) and its di erential J : E → E is de ned as We de ne the Lebesgue space L p K (R N ) composed by all measurable functions u : R N → R such that and we will state, without proof, two important results of Alves and Souto (see [[26], Lemmas 2.1 and 2.2]).
here ϕu can be expressed by the from

Proof of Theorem 1.1
To prove Lemma 3.2, we need the following results. , there is C > such that Proof. First we suppose that (VK ) holds. The proof is trivial if p = or . Now we prove that the embedding is true for p ∈ ( , ) under the assumption (VK ). For xed p ∈ ( , ), de ne m = ( −p) , and hence p = m + ( − m) . so we have that Since K(x) ∈ L ∞ (R ) and K/V ∈ L ∞ (R ), we have that Next, we suppose that (VK ) holds. Using the same argument as above, we de ne m = ( −p ) , and hence p = m + ( − m ) so that we have From (VK ) we deduce that |K(x)| |V(x)| m ∈ L ∞ (R ). It follows from the above inequality that u L p K (R ) ≤ C u E . we complete the proof.
The functional J satis es the mountain pass geometry.

Lemma 3.2. The functional J satis es the following conditions:
(i) There exist ρ and α > such that J(u) ≥ α with u E = ρ.
Proof. (i) Now, we distinguish two case.

Case 1. We suppose that (VK ) is true. For any ε > , it follows from (f ) and (f ) that there exists Cε
Thus, by ( . ) and Lemma 3.1, we get that Hence, in view of Lemma 2.3, we obtain So, taking ε = , there exists enough small u E = ρ, such that J(u) ≥ α.
Case 2. We suppose that (VK ) holds. By (f ) and (f ), there exist C ε > and C ε > such that The same as Case 1, we can take u E = ρ such that J(u) ≥ α.
(ii) For every t > we obtain From (f ), we obtain J(tu) → −∞ as t → +∞, so it is satis es (ii). We complete the proof.
Because of the appearance of the critical nonlocal term, we have to estimate the Mountain-pass value given by ( . ) carefully, To do it, we choose the extremal function . Set Vε = φUε, then thanks to the asymptotic estimates from [8], we have and for s ∈ [ , ) where S denotes the best constant for the embedding D , (R ) → L (R ), namely, We de ne Vmax := max and K min := min By the assumption (l ) we also have Proof. We now consider By Lemma 3.1, we know that, there exists tε > such that sup t≥ J(tvε) > is attained and lim t→∞ J(tvε) = −∞ for any ε > . We suppose that there exists ϱ , ϱ such that ϱ < tε < ϱ for small enough ε > . In fact, J(tε vε) = sup t≥ J(tvε), and hence dJ(tvε)/dt| t=tε = , we obtain that l(x)ϕv ε |vε| dx = .
( . ) Now, we prove that tε → +∞ as εn → + does not hold. By ( . ) which is a contradiction when tε → +∞. Similarly, we suppose that there is a sequence tε n → as εn → + . Firstly, if (VK ) holds, from (f ) and (f ), for all δ > there exists C δ > such that Choosing δ = C , it follows from ( . ) that Next, we suppose that (VK ) holds. By (f ), (f ), there is a constant C > , such that It again follows from ( . ) that We arrive at a contradiction because p > . So we complete the proof. Since < ϱ < tε < ϱ < ∞, together with the de nitions of Vmax and K min , we have We de ne By some elementary calculations, we obtain .