Existence of positive solutions for a Schrödinger-Poisson system with bounded potential and weighted functions in R

where V (x), a(x) and b(x) are positive and bounded in R, K(x) ∈ L(R) ∪ L∞(R) and K(x) ≥  in R. We will prove the existence of a positive solution (u,φ) ∈ W ,(R) × D,(R) for λ ∈ R and  < q < m < ∗, where ∗ =  is the critical exponent for the Sobolev embedding in dimension . The assumption ‘ < q < m < ’ implies that the nonlinear term f (x, u) = a(x)|u|m–u + λb(x)|u|q–u in (.) is superlinear, which is similar to those in []. Such a system, also known as the Schrödinger-Maxwell system, arises in many fields of physics. For example, the Schrödinger-Poisson system can describe the interaction of a charged particle with its own electrostatic field in quantum mechanics. The unknowns u and φ represent the wave functions associated with the particle and electric potential, and the functions V and K are, respectively, an external potential and nonnegative density


Introduction and main results
In this paper, we study the existence of positive solutions for the Schrödinger-Poisson system where V (x), a(x) and b(x) are positive and bounded in R  , K(x) ∈ L  (R  ) ∪ L ∞ (R  ) and K(x) ≥  in R  . We will prove the existence of a positive solution (u, φ) ∈ W , (R  ) × D , (R  ) for λ ∈ R and  < q < m <  * , where  * =  is the critical exponent for the Sobolev embedding in dimension . The assumption ' < q < m < ' implies that the nonlinear term f (x, u) = a(x)|u| m- u + λb(x)|u| q- u in (.) is superlinear, which is similar to those in []. Such a system, also known as the Schrödinger-Maxwell system, arises in many fields of physics. For example, the Schrödinger-Poisson system can describe the interaction of a charged particle with its own electrostatic field in quantum mechanics. The unknowns u and φ represent the wave functions associated with the particle and electric potential, and the functions V and K are, respectively, an external potential and nonnegative density charge. We refer to Benci and Fortunato [] for more details on the physical aspects. This model can also appear in semiconductor theory to describe solitary waves [].
In recent years, the Schrödinger-Poisson system [] also considered the existence of a solution for problem (.) with the potential V (x) ∈ C(R  ) satisfying inf x∈R  V (x) > -∞ and for every M > , meas({x ∈ R  |V (x) ≤ M}) < ∞. It is well known that this assumption guarantees that the embedding However, to the best of our knowledge, there are few results on problem (.) when the potential V (x) and the weighted functions a(x), b(x) are bounded in R N . In this paper, we are interested in the existence of a solution to problem (.) with V (x) satisfying Clearly, V (x) is not necessarily radial and coercive. For these assumptions, the embedding W , (R  ) → L p (R  ) is not compact. Furthermore, for problem (.), the function f (x, u) fails to satisfy the assumption uf (x, u) ≥ F(x, u). So the variational technique for problem (.) becomes more delicate. Arguing as in [, ], to preserve this compactness in some extent for our problem, we split a minimizing sequence {u n } into two parts: u n = u n + u n (n ∈ N) such that u n → u, u n →  in L m (R  , a) ∩ L q (R  , b). We will obtain a positive solution by using the Nehari manifold method.
In order to state our main results, we introduce some Sobolev spaces and norms. For p ≥ , let L p (R  ) be a usual Lebesgue space with the norm · p . Denote This norm is equivalent to the standard norm on W , (R  ) under assumption (H  ).
Let D , (R  ) be the completion of C ∞  (R  ) with respect to the norm The following Sobolev inequality [] is well known. There is a constant S >  such that for every u ∈ D , (R  ), Hence, inequality (.) holds in W , (R N ) and E. Furthermore, there exists S p >  such that for  ≤ p ≤ , It is well known that problem (.) can be reduced to a single equation with a nonlocal term, see [, ]. In fact, for every u ∈ E, we define the linear functional L u by , by the Hölder inequality and the Sobolev inequalities (.) and (.), we get Hence, by the Lax-Milgram theorem, there exists a unique φ u ∈ D , (R  ) (see [, ]) such that Moreover, φ u has the following integral expression: and φ u (x) >  in R  if u =  and K(x) ≥ . Therefore, we have from (.) that where and in the sequel, d  = K ∞ . So, it follows from (.) and (.) that there exists a constant c  >  such that Let J(u) : E → R be the energy functional associated to (.) defined by The following assumptions will be used in this paper.
(H  ) The parameters q, m and λ satisfy  < q < m <  and λ ∈ R.
Under assumptions (H  )-(H  ), it is easy to verify J ∈ C  (E, R), and for any v ∈ E, there holds Hence, if u ∈ E is a critical point of J, that is, J (u)v =  for ∀v ∈ E, then the pair (u, φ u ) is a solution of problem (.)(see [, ]). For the sake of simplicity in many cases, we just say that u ∈ E, instead of (u, φ u ) ∈ E × D , (R  ), is a weak solution of problem (.).
Our main result in this paper is as follows.

Theorem . Assume that (H  )-(H  ) hold. Then problem (.) admits at least a positive solution u ∈ E.
Open problem For K(x) ∈ L ∞ loc (R  ), does equation (.) admit a positive solution u ∈ E? Denis and Carlo in [] considered this problem with unbounded and vanishing potentials V (x). As far as we know, there is no result on the existence of positive solutions for (.) in . Hence, this case should be also an interesting topic for future research. This paper is organized as follows. In Section , we set up the variational framework and establish some lemmas, which will be used in the proof of Theorem .. In Section , we prove Theorem ..

Preliminaries
In this section, we are going to establish a series of lemmas to prove Theorem ., in which we tacitly assume that the conditions in Theorem . are satisfied. We first set up the variational framework for problem (.).
Let J(u) : E → R be the energy functional associated with problem (.) defined by (.), and its Gateaux derivative is given by (.).
Since the functional J is not bounded from below on E, a good candidate of an appropriate subset to study J is the so-called Nehari manifold for problem (.): We consider the function Lemma . The functional J is coercive and bounded from below on N . Moreover, Then, from (.) and (.), it follows that This shows that the functional J is coercive and bounded from below on N and d ≥ c  > . Then the proof of Lemma . is completed.
Let {u n } be a minimizing sequence for d in N , that is, J(u n ) → d as n → ∞ and This shows that {u n } is bounded in E and, from (.), {u n } is bounded in L m (R  , a) and L q (R  , b). Therefore, up to a subsequence, there exists u ∈ E such that as n → ∞, with some M  > . Since J(u n ) = J(|u n |), we assume u n (x) ≥  a.e. in R  for every n ≥  and thus u(x) ≥  a.e. in R  . By the weak lower semi-continuity of the norm, we get By extracting a further subsequence, if necessary, we assume By weak convergence, it is obvious that θ ∈ [, β]. First, we have the following.
Then, by the Hölder inequality and (H  ), we derive and u n → u in L q (R  , b). Hence, by the weak lower semi-continuity of the norm, we obtain Furthermore, we have from (.) that If the equality in (.) holds, then u ∈ N and the lemma is proved.
We now assume that the equality in (.) fails to hold. Let Clearly, h(t) >  for small t >  and h() < . Then there exists t ∈ (, ) such that h(t) = , and then tu ∈ N and Here and in the sequel, Relation (.) is a contradiction, and thus the proof of Lemma . is completed.
We now turn to study the value of θ . First we introduce the Sobolev space endowed with the norm where α is the positive number defined in (H  ). Consider the Schrödinger-Poisson system The functional associated with problem (.) is and the associated Nehari manifold is Finally, we define Proof Similar to the proofs of Lemmas . and ., we can obtain that N α = ∅ and d α > . Arguing in Lemma .. in [], we see that there exists the nonnegative function u  ∈ N α such that J α (u  ) = d α . On the other hand, from (H  )-(H  ), we infer Then we have As the argument of Lemma ., there exists t ∈ (, ) such that tu  ∈ N , so that where p i is given by (.). Then (.) finishes the proof of Lemma ..

Lemma . There results θ > .
Proof If θ = , then u = , which implies in particular that u n →  in L p loc (R  ) ( ≤ p < ). We first prove the following claim: Since ε is arbitrarily small, (.) implies that (.), and then where n →  as n → ∞.
On the other hand, the fact u n ∈ N shows that Since  < q < m and (.), it follows that γ n () ≥  and γ n (t) → -∞ as t → ∞. Then there exists t n ≥  such that γ n (t n ) = , and then t n u n ∈ N α , that is, where ν = lim sup n→∞ u n  E >  and β = lim n→∞ u n m m,a . Since t  ≥  and  < q < m, we have t  = . That is, t n →  as n → ∞. Therefore, it follows from (.) and (.) that Since t n → , we get γ n → . Moreover, the facts J(u n ) → d and t n →  in (.) imply that d α ≤ d. This contradicts the result in Lemma .. Therefore, we have θ >  and complete the proof of Lemma ..
In the following, we consider the case θ ∈ (, β). As in [, ], we let θ ∈ (, β) and take {u n } as a minimizing sequence for d on N , which satisfies (.) and where and in the sequel, B r = {x ∈ R  : |x| < r}, B c r = {x ∈ R  : |x| ≥ r}, n = {x ∈ R  : r n ≤ |x| < r n+ } with r n ↑ ∞ and o n () is a quantity which goes to zero as n → ∞.
Since J(u n ) = J(|u n |) in E, we assume u n ≥  in R  . Furthermore, we also consider, for every n ∈ N, a function ϕ n ∈ C ∞  (R  ) such that ()  ≤ ϕ n (x) ≤ , ∀x ∈ R  , () ϕ n (x) =  if |x| ≤ r n , ϕ n (x) =  if |x| ≥ r n+ , () |∇ϕ n (x)| ≤ C  , ∀x ∈ R  and ∀n ≥ , where C  is some positive number independent of n. Furthermore, we set Then u n , u n ≥  and u n = u n + u n in R  for every n ≥ .
Lemma . The following properties for u n , u n , u n hold: where φ u is defined by (.).
Proof The proof of properties (P  )-(P  ) is similar to that in [, ] and is omitted. Here, we prove (P  ) and (P  ). Note that u n (x) = u n (x), u n (x) =  if x ∈ B r n and u n (x) = u n (x), Thus, one sees that Clearly, which proves (P  ).
On the other hand, it follows from (.) that we have from (.) and (.) that If K ∈ L ∞ (R  ) and K ≥ , we have from (.) that = o n () and so A  n = o n (). If K ∈ L  (R  ) and K ≥ , we derive from the Hölder inequality and the Sobolev inequality that Similarly, the assumption K ∈ L  (R  ) implies that K L  ( n ) = o n () and A  n = o n (). Furthermore, since u n (x) ≥ u n (x), u n (x), we have φ u n (x) ≥ φ u n (x), φ u n (x), and then  ≤ we argue as in the proof of (.) and (.) and obtain A  n = o n (). Similarly, we get from (.) and A  n = o n () that A  n = o n () if K ∈ L  (R  ) ∪ L ∞ (R  ) and finish the proof of (P  ). Then the proof of Lemma . is completed.

Proof of Theorem 1.1
In order to prove Theorem ., we first show that the weak limit u of the minimizing sequence {u n } in Section  verifies where p i is given by (.). Let {u n } be a minimizing sequence for d on N in Section . Then, if necessary, up to a subsequence, we have u n u in E and L m (R  , a). By the weak lower semi-continuity of norms, we obtain This implies relation (.). By Lemma ., we get u ≥ , u ≡  in R  . We now prove (.). If we set Then one sees that h(t) >  for small t >  and h() < , so that there exists t ∈ (, ) such that h(t) =  and tu ∈ N and This contradiction shows that (.) cannot occur. Now we assume and choose small δ  >  such that Since u n u weakly in E, one has Moreover, it follows from (P  ) that for t > .
Obviously, γ n (t) >  for small t >  and γ n () <  for n ≥ n  . Then there exists t n ∈ (, ) such that γ n (t n ) =  and t n u n ∈ N α . Then it follows from (.) and (.) that This shows that d α ≤ d and contradicts the result in Lemma .. Therefore, (.) fails to be true and (.) is satisfied.
Since u satisfies (.) and (.), it follows that u ∈ N and so that J(u) = d and u is a minimum point for J on N .
To finish the proof of Theorem ., it is sufficient to prove that u is a critical point for the functional J(u) in E, that is, J (u)v =  for all v ∈ E, and thus J (u) =  in E * .
For every fixed v ∈ E, there exists ε >  such that u + sv ≡  for all s ∈ (-ε, ε). From Lemma ., there is t(s) ∈ R such that t(s)(u + sv) ∈ N . We will show that t = t(s) is a C  function. So, we consider the function ϕ : (-ε, ε) × R → R as ϕ(s, t) = J t(s)(u + εv) t(s)(u + sv) = t  u + sv  E + t  F(u + sv) t m u + sv m m,aλt q u + sv q q,b .
Since u ∈ N , we have Then, by the implicit function theorem, there exists a C  function t(s) such that ϕ(s, t(s)) =  and t() =  for every s ∈ (-ε  , ε  ) ⊂ (-ε, ε). This also shows that t(s) =  and t(s)(u + sv) ∈ N . Then the function h(s) = J(t(s)(u + sv)), s ∈ (-ε  , ε  ) is in C  and has a minimum at s = . Therefore, Then, from (.), it follows for every v ∈ E that J (u)v = . Thus J (u) =  in E * . So, u is a critical point for J, and then u is a weak solution of problem (.) in E, that is, u is a solution of (.). Since J(u) = J(|u|) = d > , we can assume u ≥  a.e. in R  . Furthermore, the application of maximum principle in [] yields u(x) >  in R  . Then the proof of Theorem . is finished.