Multiple solutions for a quasilinear Schrödinger–Poisson system

where V ,K :R3 →R and g :R3 ×R →R are continuous functions; g is of subcritical growth and has some monotonicity properties. The purpose of this paper is to find the ground state solution of (0.1), i.e., a nontrivial solution with the least possible energy by taking advantage of the generalized Nehari manifold approach, which was proposed by Szulkin and Weth. Furthermore, infinitely many geometrically distinct solutions are gained while g is odd in u.


Introduction
In this article, we consider the following quasilinear Schrödinger-Poisson system: where V , K : R 3 → R and g : R 3 × R → R are continuous functions.
The quasilinear Schrödinger-Poisson system had been introduced in [4,20], which is a quantum mechanical model of extremely small devices in semiconductor nanostructures taking into account the quantum structure and the longitudinal field oscillations during the beam propagation.
The Schrödinger-Poisson system can be written as follows: There were many different growth conditions on the nonlinearity g(x, u), such as subcritical growth [6,21,31,33,36], or the critical exponent growth [32,34]. Moreover, many meaningful results have been obtained. For example, radial solutions [8,11,22], ground state solutions [2,3], and also semiclassical solutions [12,18,19]. The quasilinear Schrödinger equation has been accepted as a model of several physical phenomena in [23], and for the results about this quasilinear equation the reader can be referred to [1,5,9,10,15,30]. At present, there are relatively few existing results about system (1.1), so it could be of interest to pay close attention to our discussion. When K(x) = 1 in system (1.1), the authors of [7] took into account the system together with a 4-Laplacian operator, hence the existence of sign-changing solution with precisely two nodal domains was derived by using the approximation technique. When the system is considered with asymptotically linear f (t) with respect to t at infinity, the existence and asymptotic behavior of the ground state were studied in [13]. Figueiredo and Siciliano [16,17] paid close attention to two different systems with parameter ε and critical growth, and obtained the existence of solutions for the former system in R 3 . They also proved asymptotic behavior of solutions whenever ε → 0. Similar results were also achieved for the latter system in a bounded domain in R 2 . The authors in [24] considered a system with radial potentials and discontinuous nonlinearity, and then obtained the multiplicity results of radial solutions by nonsmooth critical point theory. By utilizing Ekeland's variational principle, the authors of [25] obtained the existence of the ground state solution by seeking the solution of a new system, which is equivalent to system (1.1). By applying the mountain pass theorem and the concentrationcompactness principle, the existence of a solution to problem (1.1) was established in [35] when the asymptotic periodicity of potentials V , K and nonlinearity g were considered.
To the best of our knowledge, no one made use of the generalized manifold method to show the existence of solutions of this quasilinear Schrödinger-Poisson system. On the basis of the existing results, our aim is to study the existence of a ground state solution and of infinitely many solutions.
Setting G(x, u) = u 0 g(x, s) ds, we suppose that V , K and g satisfy the following assumptions: for some a > 0 and 4 < p < 12; u 3 is positive for u = 0, nonincreasing on (-∞, 0), and nondecreasing on (0, +∞); (g 4 ) G(x,u) u 4 → ∞ uniformly in x as |u| → ∞. Let * denote the action of Z 3 on H 1 (R 3 ) given by We note that if u 0 is a solution of (1.1), then so are all elements of the orbit of (u 0 , φ 0 ) under the action of Z 3 , and the so-called orbit of (u 0 , φ 0 ) is denoted as Two solutions (u 1 , φ 1 ) and (u 2 , φ 2 ) of (1.1) are said to be geometrically distinct if O(u 1 , φ 1 ) and O(u 2 , φ 2 ) are disjoint. Now we state our main results.

Preliminary results
In this section, we introduce the variational framework associated with problem (1.1). It follows from (V ) that we can define a new norm which is equivalent to the usual norm of H 1 (R 3 ). The usual norm in the Lebesgue space L p (R 3 ) is denoted by |u| p . Then there exists r p > 0 such |u| p ≤ r p u for p ∈ [1,6]. Also u D 1,2 = R 3 (|∇u| 2 ) 1/2 denotes the norm of D 1,2 (R 3 ). It is more convenient for our purposes than the standard one and will be used henceforth. For a functional F, we put As far as we know, system (1.1) can be easily transformed into a single nonlinear Schrödinger equation with a nonlocal term. Briefly, the Poisson equation is solved by using the Lax-Milgram theorem, so for all u ∈ H 1 (R 3 ), a unique φ u ∈ D 1,2 (R 3 ) is gained, such thatφ = K(x)u 2 and, when inserted into the first equation, it gives Actually, for each u ∈ H 1 (R 3 ), we define an operator T u on D 1,2 (R 3 ) by Hölder inequality and the fact K ∈ L 2 (R 3 ) yield that there is a constant C > 0 such that for every v ∈ D 1,2 (R 3 ), Hence, by the Riesz representation theorem, there exists a unique φ u ∈ D 1,2 (R 3 ) such that Thus φ u is a weak solution ofφ u = K(x)u 2 and can be represented by Moreover, it is obvious that Also there is Let us now define the operator : then has the following properties [11]: is continuous and maps bounded sets into bounded sets; We observe that (1.1) is formally the Euler-Lagrange equation associated with the energy functional From the variational point of view, the first difficulty associated with the problem (1.1) is finding an appropriate function space where the functional J is well defined. In order to avoid the difficulty which is caused by the quasilinear term, we take advantage of the change of variable introduced by [23], that is, we consider having the following properties, which have been proved in [9,14].

Lemma 2.2 The function f satisfies the following properties:
(1) f is uniquely defined, C ∞ , and invertible; So, after the change of variables, from J we obtain the following functional: which is well defined in H 1 (R 3 ) and belongs to C 1 under the hypotheses (V ), (K), (g 1 ), and (g 2 ). Moreover, the critical points of I are the weak solutions of the problem that is, . It has been shown in [9] that if v ∈ H 1 (R 3 ) is a critical point of the functional I, then u = f (v) ∈ H 1 (R 3 ) and u is a solution of (1.1).
We also observe that, in order to obtain a nonnegative solution for (1.1), we set g(x, s) = 0 for all x ∈ R 3 , s < 0. Indeed, let v be a critical point of I.
Hence we may conclude that v -= 0 almost everywhere in R 3 and, therefore, v = v + ≥ 0.
As u = f (v), we conclude that u is a nonnegative solution for the problem (1.1). Let Recall that M is called the Nehari manifold. We do not know whether M is of class C 1 under our assumptions and therefore we cannot use the minimax theory directly on M.
To overcome this difficulty, we employ an argument developed in [27,28].
Using Lemma 2.2, (3) and (7), and (2.2), then we can easily gain that h(t) → -∞ as t → ∞ according to (2.1), Lemma 2.2(5), (g 4 ), and Fatou's lemma. Therefore, max t>0 h(t) is achieved at some t u = t(u) > 0, so that h (t u ) = 0 and then t u u ∈ M. It remains to show the uniqueness of t u . Suppose by contradiction that there exists a t 1 > 0 with t u = t 1 such that h (t 1 this implies that contrary to (g 3 ).
Proof (1) According to [15], R 3 |∇u| 2 + R 3 V (x)f 2 (u) ≥ C u 2 whenever u ≤ ρ. By (3.1) and Lemma 2.2, (3) and (7), we have for sufficiently small ε, and then inf S ρ I > 0 is obtained when ρ is small enough. The inequality inf M I ≥ inf S ρ I is a consequence of Lemma 3.1 since for every u ∈ M there is s > 0 such that su ∈ S ρ and I(t u u) ≥ I(su).
(3) Arguing indirectly, let (u n ) ⊂ M be a sequence such that u n → ∞ and I(u n ) ≤ d for some d. Set v n = u n / u n . Then v n = 1 and, passing to a subsequence, v n v in H 1 (R 3 ) and v n → v a.e. in R 3 . Suppose where B 1 (x) is the ball in R 3 with center x and radius 1, then it follows that v n → 0 in L r (R 3 ) for 2 < r < 6 by Lions' lemma (cf. [29,Lemma 1.21]). Using (3.1) and Lemma 2.2, (3) and (7), we see that G(x, f (sv n )) → 0 for all s ∈ R. Hence by Lemma 2.2(9), This yields a contradiction if we choose a sufficiently large s. Hence (3.3) is not true and, since I and M are invariant under the action of Z 3 , after a suitable Z 3 translation it follows that v n → v = 0 in L 2 loc (Z 3 ). Since |u n (x)| → ∞ if v(x) = 0, it follows from Lemma 2.2(5), (g 4 ), and Fatou's lemma that and also, by Lemma 2.2, (3) and (7), as well as (2.2), thus the proof is completed.

Lemma 3.3 If V is a compact subset of H 1 (R 3 ) \ {0}, then there exists R > 0 such that I
Proof We may assume without loss of generality that V ⊂ S. Arguing by contradiction, suppose there exist u n ∈ V and w n = m(u n ) = t n u n such that I(w n ) ≥ 0 and t n → ∞ as n → ∞. Passing to a subsequence, there is u ∈ H 1 (R 3 ) with u = 1 such that u n → u ∈ S. Since |w n (x)| → ∞ if u(x) = 0, then by (2.2) and Lemma 2.2(7) it follows that so by (g 4 ), Lemma 2.2(5), and Fatou's lemma we get that Therefore, by Lemma 2.2(3), Recall that S is the unit sphere in H 1 (R 3 ) and define the mapping m : S → M by setting where t w is as in Lemma 3.1. Note that m(w) = t w . Lemmas 3.4 and 3.5 below are taken from [28] (see Proposition 8 and Corollary 10 there). That the hypotheses in [28] are satisfied is a consequence of Lemmas 3.1, 3.2 and 3.3 above. Indeed, if h(t) = I(tw) and w ∈ S, then h (t) > 0 for 0 < t < t w and h (t) < 0 for t > t w by Lemma 3.1, t w ≥ δ > 0 by Lemma 3.2, and t w ≤ R for w ∈ V ⊂ S by Lemma 3.3.

Lemma 3.4 The mapping m is continuous. Moreover, the mapping m is a homeomorphism between S and M, and the inverse of m is given by m
We shall consider the functional : S → R given by (w) = I m(w) . (2) If (w n ) is a Palais-Smale sequence for , then (m(w n )) is a Palais-Smale sequence for I. If (u n ) ⊂ M is a bounded Palais-Smale sequence for I, then (m -1 (u n )) is a Palais-Smale sequence for .

(3) w is a critical point of if and only if m(w) is a nontrivial critical point of I.
Moreover, the corresponding values of and I coincide and inf S = inf M I. (4) If I is even, then so is .
If u n is bounded in H 1 (R 3 ), then passing to a subsequence gives that u n u in H 1 (R 3 ) and u n → u a.e. in R 3 . Then by (2.1) it follows that φ u n is bounded in D 1,2 (R 3 ), and also φ u n is bounded in L 6 (R 3 ). Then by Lemma 2.2(3) we obtain that φ f (u n ) is bounded in L 6 (R 3 u 0 . Since u n → u a.e. in R 3 and due to the uniqueness of limit, we have φ f (u n ) φ f (u n ) in L 6 (R 3 ). Moreover, by Lemma 2.2, (8) and (3), |u n | 12 5 ≤ C 8 u n 12 5 , so f (u n )f (u n )u n ∈ L 6 5 (R 3 ), and then I 1 → 0. Moreover, I 2 → 0, I 3 → 0, and I 4 → 0 can be obtained in a similar way.
We also can use the same method, which was used in (1) to prove I 5 → 0 and I 6 → 0. And then the proof is completed. Now we are in a position to prove Theorem 1.1.
Proof of Theorem 1. 1 We take advantage of an argument in [27]. Let c = inf M I. It follows from Lemma 3.2(1) that c > 0. Moreover, if u 0 ∈ M satisfies I(u 0 ) = c, then m -1 (u 0 ) ∈ S is a minimizer of and therefore a critical point of , thus, by Lemma 3.5(3), u 0 is a critical point of I. It remains to show that there exists a minimizer u of I| M . By Ekeland's variational principle [29], there exists a sequence (w n ) ⊂ S with (w n ) → c and (w n ) → 0 as n → ∞. Set u n = m(w n ), then, from Lemma 3.5(2), we conclude that I(u n ) → c and I (u n ) → 0 as n → ∞. Obviously, (u n ) is bounded by Lemma 3.2(3). Therefore u n u after passing to a subsequence. Suppose Furthermore, by Lemma 2.2, (6), (8) and (9), According to (3.1) and using Lemma 2.2, (3) and (7), then which means that R 3 g(x, f (u n ))f (u n ) → 0. Moreover, which yields a contradiction, so that (3.4) cannot hold. Thus, after a suitable Z 3translation, up to a subsequence we have u n u = 0 and it is well known that I (u) = 0. Accordingly, u ∈ M, and also that I(u) ≥ c. In order to complete the proof, we only need to show that I(u) ≤ c. Note that From Lemma 2.2(8) and Fatou's lemma, we have V (x) f 2 (u)f (u)f (u)u and, by Lemma 2.2(6), (3.2), and Fatou's lemma, In the following, we devote ourselves to the proof of the multiplicity result of Theorem 1.2. In addition to the assumptions (V ), (K), and (g 1 )-(g 4 ), we also suppose that g is h(w) = η(e(w), w) is odd and continuous, so it follows from the properties of the genus and the definition of c k that γ ( d+ε \ U) ≤ γ ( d-ε ) ≤ k -1, which implies γ d+ε ≤ γ (U) + k -1 = γ (K d ) + k -1. (3.10) If γ (K d ) = 0, then from above we have γ ( d+ε ) ≤ k -1, contrary to the definition of c k . Therefore, γ (K d ) = 1 and so K d = ∅. Suppose c k+1 = c k , then by (3.10), the definition of d = c k and of c k+1 we deduce that γ (K d ) ≥ 2, contrary to the fact that γ (K d ) = 1. Hence c k < c k+1 and so (3.9) is obtained. The proof is completed.