Multiple nontrivial solutions for a nonhomogeneous Schrödinger – Poisson system in R 3

In this paper, we study the following Schrödinger–Poisson system { −∆u + V(x)u + φu = f (x, u) + g(x), x ∈ R3, −∆φ = u2, x ∈ R3. Under appropriate assumptions on V, f and g, using the Mountain Pass Theorem and the Ekeland’s variational principle, we establish two existence theorems to ensure that the above system has at least two different solutions. Recent results from the literature are extended and improved.


Introduction and main results
In this paper, we consider the following nonlinear Schrödinger-Poisson system where V ∈ C R 3 , R , f ∈ C R 3 × R, R and the conditions on g will be given later.System (1.1) is also called Schrödinger-Maxwell system, arises in an interesting physical context.In fact, according to a classical model, the interaction of a charge particle with an electromagnetic field can be described by coupling the nonlinear Schrödinger's and Poisson's equations.For more information on the physical relevance of the Schrödinger-Poisson system, we refer the readers to the papers [3,23] and the references therein.
It is worth pointing out that the combination of ( f 3 )-( f 4 ) implies that the rang of p in condition ( f 1 ) should be 4 < p < 6.In fact, for any where, c This contradicts (1.3).Thus, 4 < p < 6. Inspired by the above facts, in the present paper we shall consider the nonhomogeneous Schrödinger-Poisson system, and we are interested in looking for multiple solutions for the problem (1.1).Under much more relaxed assumptions on the nonlinearity f and the potential function V, using some special proof techniques especially the verification of the boundedness of Palais-Smale sequence, new results on the existence of multiple nontrivial solutions for the system (1.1) are obtained, which extend and sharply improve some recent results in the literature.In order to state the main results of this paper, we make the following assumptions.  where (H 4 ) There exist c 3 > 0 and L > 0 such that 4F(x, t) ≤ f (x, t)t + c 3 t 2 , for a.e.x ∈ R 3 and ∀ |t| ≥ L.
Now, we are ready to state the main results of this paper as follows.
Theorem 1.2.Assume that (V) and (H 1 )-(H 5 ) hold.Then, there exists m 0 > 0 such that for any g ∈ L p (R 3 ) with g p ≤ m 0 , the system (1.1) possesses at least two different nontrivial solutions, one is negative energy solution, and the other is positive energy solution.
The other aim of this paper is to study the existence of at least two different nontrivial solutions for problem (1.1) involving a concave-convex nonlinearity.We also consider the effect of the parameter λ and the perturbation term g on the existence of solutions.
Moreover, there exists a nonempty bounded domain Ω ⊂ R 3 such that h 2 > 0 in Ω.
Then there exist λ 0 , m 0 > 0 such that for all λ ∈ (0, λ 0 ), the system (1.1) possesses at least two different nontrivial solutions whenever g 2 ≤ m 0 , one is negative energy solution, and the other is positive energy solution.
Obviously, the condition (H 4 ) implies the condition (H 4 ), so we have the following corollary.
Remark 1.5.Since the problem (1.1) is defined in the whole space R 3 , the main difficulty of this problem is the lack of compactness for Sobolev embedding theorem.To overcome this difficulty, the condition (V), which was firstly introduced by Bartsch et al. [4], is always assumed to preserve the compactness of the embedding of the working space.Furthermore, condition (V) is weaker than condition (V 0 ), and there are functions V(x) satisfying (V) but not satisfying (V 0 ), see for example Remark 2 in [33].
(1) Theorem 1.2 sharply improves Theorem 1.1.If fact, from Remark 3. in [33], we know that the condition (H 1 ) is much weaker than the combination of ( f 1 ) and ( f 2 ), and conditions ( The condition (H 2 ) which gives the behaviour of f (x, u)/u for u near to the origin, is very essential for obtain the positive energy solution in Theorem 1.2.Moreover, it seems to be nearly optimal for obtain a such existence result.
(3) As a function f satisfying the assumptions (H 1 )-(H 4 ), one can take where 0 < ν < µ * 2 (µ * is given by (H 2 )).A straightforward computation deduces that Hence, it is easy to check that f satisfies the assumptions (H 1 )-(H 4 ).However, it does not satisfy the assumptions of Theorem 1.1.In fact, we have lim t→0 f (x,t) t = 2ν > 0 uniformly for x ∈ R 3 , which implies that f does not satisfy the condition ( f 2 ).Moreover, for any µ > 4, we have which shows that the condition ( f 3 ) is not satisfying for our choice.
Remark 1.7.The assumptions of Theorems 1.2 and 1.3 can be used to deal with the existence of nontrivial solutions for the following nonhomogeneous Kirchhoff-type equations where a > 0, b ≥ 0 are constants.So, the conclusions of Theorems 1.2 and 1.3 still hold for the above problem.
The paper is organized as follows.In Section 2, we present some preliminary results.Section 3 is devoted to the proof of Theorems 1.2 and 1.3.

Preliminaries
In the following, we will introduce the variational setting for Problem (1.1).In the sequel, we denote by • p the usual norm of the space L p R 3 , c i , C i or C stand for different positive constants.
As usual, for 1 ≤ p < +∞, we let and u ∞ := ess sup with the inner product and norm Define our working space Then E is a Hilbert space equipped with the inner product and norm Let D 1,2 (R 3 ) be the completion of C ∞ 0 (R 3 ) with respect to the norm Then, the embedding D 1,2 (R 3 ) → L 6 (R 3 ) is continuous (see for instance [29]).Since the embedding
Recall that µ ∈ R is called an eigenvalue of the operator −∆ + V(x) provided there exists a nontrivial weak solution u 0 of the equation: Lemma 2.2.Assume that (V) holds.Then µ * is an eigenvalue of the operator −∆ + V(x) and there exists a corresponding eigenfunction ϕ 1 with ϕ 1 > 0 for all x ∈ R 3 .
Proof.The proof of this lemma is almost the same to the one of Lemma 2.3 in [13].So we omit it here.
For every u ∈ H 1 (R 3 ), by the Lax-Milgram theorem, we know that there exists a unique Furthermore, φ u has the following integral expression From (2.1), for any u ∈ E, using the Hölder inequality we obtain Therefore By (2.4), (2.5) and the Sobolev inequality, we obtain 1 4π Moreover, φ u has the following properties (for a proof, see [6,21]).
Lemma 2.3.For u ∈ E we have Now, we define the energy functional J : E → R associated with problem (1.1) by Therefore, combining (2.5), (2.6), (H 1 )-(H 2 ) and Lemma 2.1, J is well defined and (2.8) Moreover, if u ∈ E is a critical point of J, then the pair (u, φ u ) is a solution of system (1.1).
Recall that a sequence {u n } ⊂ E is said to be a Palais-Smale sequence at the level c ∈ R ((PS) c -sequence for short) if J(u n ) → c and J (u n ) → 0, J is said to satisfy the Palais-Smale condition at the level c ((PS) c -condition for short) if any (PS) c -sequence has a convergent subsequence.
Then J has at least a critical value c ≥ α.
On the other hand, the following Ekeland's variational principle is the main tool to obtain the negative energy solution for problem (1.1) Proposition 2.5 ([19,Theorem 4.1]).Let M be a complete metric space with metric d and let J : M → (−∞, +∞] be a lower semicontinuous function, bounded from below and not identical to +∞.Let ε > 0 be given and u ∈ M be such that Then, there exists v ∈ M such that and for each w ∈ M, one has J(v) ≤ J(w) + εd(v, w).
We also need the following auxiliary result, see [27].

Proof of main results
In this section we shall prove Theorems 1.2 and 1.3.We first prove some lemmas, which are crucial to prove our main results.
Proof.Let {u n } ⊂ E be such that We first show that {u n } is bounded in E. Otherwise, set v n = u n u n , then v n = 1 and where L > 0 is given by (H 4 ).Combining the above inequality with (H 4 ), we conclude that there exists c 8 > 0 such that By (H 5 ), (2.7), (2.8), (3.5), (3.7) and the Hölder inequality, without loss of generality, we may assume that for all n ∈ N, we have Therefore, for sufficiently large n such that 4(c+1) ≤ 1 2 , we then get Consequently, we conclude that v n 2 > 0. ( Let Hence, A n ⊆ R 3 \ Ω n for n ∈ N large enough.It follows from (H 5 ) and the Hölder inequality that for any β ∈ (1, 6), one has since u n → ∞ as n → ∞.By (H 1 ), (H 3 ), (2.1), (2.6), (3.5), (3.6), (3.8), (3.9) and Fatou's lemma, we have This is an obvious contradiction.Hence {u n } ⊂ E is bounded.So, up to a subsequence we may assume that u n u 0 weakly in E. By Lemma 2.1, u n → u 0 strongly in L s R 3 for 2 ≤ s < 6 and u n (x) → u 0 (x) a.e. on R 3 .It follows from (2.7) and (2.8) that Let us take r = q = 1 in Lemma 2.6 and combine with u n → u 0 strongly in L s R 3 for 2 ≤ s < 6, to get Furthermore, from Lemma 2.3 (ii), we have that R 3 (φ u n u n − φ u 0 u 0 )(u n − u 0 )dx → 0. Consequently, u n → u 0 in E. This completes the proof.
Proof of Theorem 1.2.The proof is divided in two steps, the first one for the negative energy solution, the second one for the positive energy solution.
Step 1.By using Ekeland's variational principle, we first show that there exists a function u 0 ∈ E such that J (u 0 ) = 0 and J(u 0 ) < 0. By (3.3) fixing M > 0 a constant C M > 0 exists such that Since g ∈ L p (R 3 ) and g ≡ 0, we may choose a function v ∈ E such that Therefore, for t > 0 small enough, which implies that inf{J(u) : where ρ > 0 is given by Lemma 3.1, and B ρ = {u ∈ E : u ≤ ρ}.On the other hand, by (3.2), one has which implies that J is bounded below in B ρ .Thus, we obtain By Ekeland's variational principle, there exists a sequence {u n } ⊂ B ρ such that Then, following the idea of [11] (see pp. 534-535), we can show that {u n } is a bounded Palais-Smale sequence of J. Therefore, by Lemma 3.3, {u n } has a strongly convergent subsequence, still denoted by {u n } and u n → u 0 ∈ B ρ as n → ∞.Hence, we conclude that there exists u 0 ∈ E such that J(u 0 ) = inf u∈B ρ J(u) = c 0 < 0 and J (u 0 ) = 0, this completes the Step 1.
Step 2. Now, we show that there exists a function u 0 ∈ E such that J(u 0 ) = c 0 > 0 and J (u 0 ) = 0 by means of the Mountain Pass Theorem.Obviously, J ∈ C 1 (E, R) and J(0) = 0.By Lemmas 3.1 and 3.2, the functional J satisfies the geometric property of the mountain pass theorem whenever g p ≤ m 0 .Lemma 3.3 implies that J satisfies the (PS)-condition.Therefore, applying Proposition 2.4, we deduce that there exists u 0 ∈ E such that J(u 0 ) = c 0 ≥ α > 0 and J (u 0 ) = 0, we complete the Step 2. Therefore, by the above two steps the proof of Theorem 1.2 is completed.
Next, we will give the proof of Theorem 1.3.Under the assumption (H 6 ), we can easily find that the energy functional associated to problem (1.1) is of class C 1 on E and for any v ∈ E, we have Lemma 3.4.Suppose that the assumptions (V) and (H 6 ) are satisfied.Then, there exist ρ, α and m 0 > 0 such that J(u) ≥ α whenever u = ρ and g 2 < m 0 .
Because 1 < σ < 2 and p > 4, we deduce that {u n } is bounded in E. Therefore, there exists u ∈ E such that, up to a subsequence, we have u n u weakly in E, u n → u strongly in L s R 3 for 2 ≤ s < 6 and u n (x) → u(x) a.e. on R 3 .Similar to the proof of Lemma 3.3 (see (3.11)), in order to prove that u n → u strongly in E, it sufficient to show that Since u n → u strongly in L s R 3 for 2 ≤ s < 6, the Hölder inequality implies that Therefore, J satisfies the (PS)-condition.
Proof of Theorem 1.3.Similar to the proof of Theorem 1.2, we also divide the proof into two steps.
Step 1.As the proof of Step 1 in Theorem 1.2, we first prove the existence of negative energy solution via Ekeland's variational principle (cf.Proposition 2.5).Since g ∈ L 2 (R 3 ) and g ≡ 0, we can choose a function v ∈ E such that R 3 g(x)v(x)dx > 0.