Existence of Multiple Positive Solutions for Choquard Equation with Perturbation

Tao Xie, Lu Xiao, and Jun Wang 1School of Management, Jiangsu University, Zhenjiang, Jiangsu 212013, China 2Faculty of Science, Jiangsu University, Zhenjiang, Jiangsu 212013, China Correspondence should be addressed to Lu Xiao; hnlulu@126.com Received 7 May 2015; Revised 17 July 2015; Accepted 13 September 2015 Academic Editor: Kamil Brádler Copyright © 2015 Tao Xie et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. This paper is concerned with the following Choquard equation with perturbation: −Δu + V(x)u = (1/|x|α ∗ |u|p)|u|p−2u + g(x),

This kind of (1) arises in various physical contexts, especially in the case where  = 3,  = 2,  = 2, and  = 0. Then (1) becomes It is called the stationary nonlinear Choquard equation or the nonlinear Schrödinger-Newton equation.In general, many mathematicians are concerned with the positive solitary solutions of the following nonlinear generalized Choquard equation: where the powers  ≥ 2 and  ∈ (0, ).In order to obtain the solitary solutions of (3), we set (, ) =   () ( > 0 is a constant) in (3) and get the stationary equation of (1) without perturbation, where () = () − .
In 1954, paper [1] proposed model (2) to the description of the quantum theory of a polaron.Later, (2) was proposed by Choquard in 1976 as an approximation to Hartree-Fock theory for one component plasma [2].In the 1990s the same equation reemerged as a model of self-gravitating matter [3,4] and is known in that context as the Schrödinger-Newton equation.In recent years, many papers are concerned with the existence of solutions of (3).Lieb [2] proved the existence and uniqueness of the ground state to (2).Lions [5] obtained the existence of a sequence of radially symmetric solutions for (2) by using variational methods.Papers [6,7] proved the existence of multibump solutions of (2).Paper [8] proved that 2 Advances in Mathematical Physics every positive solution of ( 2) is radially symmetric and monotone decreasing about some point by using moving plane methods.Furthermore, the authors obtained the uniqueness of positive solutions for (2).Clapp and Salazar [9] proved the existence of positive and sign changing solutions of (2) when R 3 and  are replaced by an exterior bounded domain Ω and (), respectively.Moroz and Van Schaftingen [10] showed the regularity, positivity, and radial symmetry of the ground states for the optimal range of parameters, and they also obtained decay asymptotic at infinity for these ground states.The more general system (3) was considered in [11].Moroz and Van Schaftingen [12] obtained the nonexistence and optimal decay of supersolutions of (3).Cingolani and Secchi [13] considered the existences of ground states for the pseudorelativistic Hartree equation.For semiclassical cases, the existence of multiple semiclassical solutions was considered in [14].Paper [15] considered the existence of semiclassical regime of standing wave solutions of a Schrödinger equation in presence of nonconstant electric and magnetic potentials.Cingolani and Secchi [16] studied the semiclassical limit for the pseudorelativistic Hartree equation.Under the assumptions on the decay of , paper [17] proved the existence of positive solutions by using variational methods and nonlocal penalization technique.
Motivated by the works we mentioned above, in this paper we study the existence of multiple solutions to the nonlinear Choquard equation with perturbation.This kind of problems is often referred to as being nonlocal because of the appearance of the term ∫ R  ∫ R  (|()|  |()|  /|−|  )  in the energy functional.This leads to the fact that (1) is no longer a pointwise identity.The main difficulties when dealing with this problem lie in the presence of the nonlocal term and the lack of compactness due to the unboundedness of the domain R  .Under some conditions on , in the present paper we recover the compactness and find two nontrivial solutions of (1) by using variational methods.
In what follows, we assume that  ∈ (R  , R + ) and satisfies the following condition: A solution is called a ground state solution (or positive ground state solution) if its energy is minimal among all the nontrivial solutions (or all the nontrivial positive solutions) of (1).A bound state solution refers to limited-energy solution.
Then, we have the following main results.

Variational Setting
Throughout the paper, we use the following notations: (iii) Let  and   be some positive numbers.
The main purpose of this section is to establish the variational setting for problem (1).We first recall the following classical Hardy-Littlewood-Sobolev inequality (see [20,Theorem 4.3]).
In order to prove Theorem 1 we will constrain the functional  on the set Usually, this set is called Nehari manifold.It is well-known that critical points of  lie in the Nehari manifold.Denote Φ() = ⟨  (), ⟩.Thus, we know that In order to prove the existence of multiple nontrivial solutions for (1), we will divide the Nehari manifold N into the following three parts: Obviously, only N 0 contains the element 0. Furthermore, it is easy to see that N + ∪N 0 and N − ∪N 0 are both closed subsets of .
Next we will give some explanation for the partition of Nehari manifolds N. Set We define the fibering map Thus, Obviously,  ∈ N with  > 0 if and only if   () = 0.
It is well-known that if the function () has unique global maximum point, then the set N is homotopic to unit ball of .Moreover, the set N is a natural constraint for the functional .This means that if the infimum is attained by  ∈ N, then  is a solution of (1).However, in our situation, the global maximum point of  is not unique.This leads us to partition the set N according to the critical points of .This kind of idea was first introduced by Tarantello in [21].Later, many mathematicians apply this idea to study other problems; for instance, see [22][23][24] and the references therein.
Now we are ready to study the properties of sets N ± and N 0 .
Since  = 1 and  lies on the unit sphere of , we infer from Lemma 3 that  has upper bound.So there exists  2 > 0 such that .

Proof of Theorem 1
In this section we are going to give the proof of the main results.Before doing this we should study the properties for the minimizing sequences for the functional .In the whole paper, we say lim  → ∞   (  ) = 0 means that lim  → ∞ ‖  (  )‖ = 0. Lemma 6.Under the assumptions of Theorem 1, there exists a sequence {  } ⊂ N + such that (  ) →  + and   (  ) → 0 as  → ∞.

Solutions for the Choquard Equation with General Nonlinearity
In this section we will look for the positive solutions for the following Choquard equation with general nonlinearity: where () = ∫  0 ().Since we only care about the existence of positive solutions, in what follows, we assume that  ∈  1 (R + , R) verifies the following conditions.