On a class of nonlocal nonlinear Schrödinger equations with potential well

Abstract: In this paper we investigate the existence, multiplicity and asymptotic behavior of positive solution for the nonlocal nonlinear Schrödinger equations. We exploiting the relationship between the Nehari manifold and eigenvalue problems to discuss how the Nehari manifold changes as parameters μ, λ changes and show how existence, multiplicity and asymptotic results for positive solutions of the equation are linked to properties of the manifold.


Introduction
In this paper we are concerned with the existence and multiplicity of positive solutions of the nonlocal nonlinear Schrödinger equation where N ≥ , N+α N ≤ p < N N− and Iα is the Riesz potential of order < α < min N, * ( * = N N− ) on the Euclidean space R N , de ned for each point x ∈ R N \{ } by with Γ being the Euler gamma function. Throughout this paper, we assume that the parameters µ, λ > and the functions V µ,λ := µg − λa and f satisfy the following conditions: (V )g is a nonnegative continuous function on R N ; (V )there exists c > such that the set {g < c} := x ∈ R N | g (x) < c is nonempty and has nite measure; (V )Ω = int x ∈ R N | g (x) = is nonempty bounded domain and has a smooth boundary with Ω = x ∈ R N | g (x) = ; (V )a ∈ L N/ R N ∩ L ∞ R N and x ∈ Ω : a (x) > > ; (F )f ∈ L ∞ R N and x ∈ Ω : f (x) > > .
Remark 1.1. By condition (V ) , the set x ∈ Ω : a (x) > has positive Lebesgue measure, we can assume that λ (a Ω ) denote the positive principal eigenvalue of the problem −∆u(x) = λa Ω (x)u(x) for x ∈ Ω; u(x) = for x ∈ ∂Ω, (1.1) where a Ω is a restriction of a on Ω Clearly, λ (a Ω ) has a corresponding positive principal eigenfunction ϕ .
In recent years, nonlinear Schrödinger type equation has been widely studied under variant assumptions on potential g and weight function f . Most of the literature has focused on the equation for g being a positive potential and f being a positive weight function with satis es the some assumptions of in nite limits. Moreover, the conditions (V ) − (V ) imply that µg represents a potential well whose depth is controlled by µ. µg is called a steep potential well if µ is su ciently large and one expects to nd solutions which localize near its bottom Ω. This problem has found much interest after being rst introduced by Bartch and Wang [11] in the study of the existence of positive solutions for nonlinear Schrödinger equations and has been attracting much attention, see [3,9,10,38,43] and the references therein. Later, the steep potential well is introduced to the study of some other types of nonlinear di erential equations by some researchers, such as nonlocal nonlinear elliptic equations [18,24,33,34,45,46]. When N = and the nonlocal nonlinear term Iα * u p |u| p− u = I * u u for α = p = . Then Eq. (P µ,λ ) is the one type of the following nonlocal nonlinear Schrödinger equation: where < p < and the parameter σ > . It is easy to know that u is a solution of Eq. ( . ) if and only if (u, ϕ) is a solution of the following equation: It is well known that Eq. ( . ) is called the Schrödinger-Poisson system, which was rst introduced in [7] as a physical model describing a charged wave interacting with its own electrostatic eld. Eq. ( . ) also appears in the electromagnetic eld, semiconductor theory, nonlinear optics and plasma physics. Due to the important applications in physics, Eq. ( . ) has been widely studied via modern variational methods under various hypotheses on the potential function and the nonlinearity; see [2,4,16,23,30,31,[35][36][37] and the references therein. More precisely, Ruiz [30] obtained the existence, nonexistence and multiplicity of radial positive solutions for Eq. ( . ) with V = f ≡ . It turn out that p = is a critical value for the existence of nontrivial solutions. Ruiz's approach is based on minimizing the energy functional I associated with Eq. ( . ) on a certain manifold that is the Nehari-Pohozaev manifold: where H r (R ) consists of radially symmetric functions in H (R ) and Q(u) = is derived by subtracting the Pohozaev identity of equations ( . ) from the equation I(u), u = . They proved that when < p < / and for σ is su ciently small, Eq. ( . ) has two positive radial solutions v , v with In recent years, many authors have been studying such topics (existence of two positive solutions which one of the negative energy), for example, Chen [14], Huang et al. [21,22] and Shen and Han [32], consider the following Schrödinger-Poisson system where < p < , l ∈ L R , f ∈ C R changes sign in R and lim |x|→∞ f (x) = f∞ < . They proved that system ( . ) has two positive solutions which one of the negative energy for λ > λ (h), where λ (h) is the rst eigenvalue of −∆ + id in H R with nonnegative weight function h ∈ L / R . Very recently, it has proven in [36] that the problem ( . ) admits a positive solution when V ≡ , < p ≤ , and σ belongs to a certain interval. To this end, the authors introduced the ltration of the Nehari manifold N, that is N(c) = {u ∈ N : I(u) < c}, and showed that this set N(c) under the given assumptions is the union of two disjoint nonempty sets, namely, which are both C sub-manifolds of N(c) and natural constraints of I. Moreover, N ( ) is bounded such that I is coercive and bounded below on it, whereas I is unbounded below on N ( ) . Moreover, they use the argument of concentration compactness principle to obtain a minimizer of I on N ( ) , which is a critical point of I. Actually the authors also established N ( ) may not contain any non-zero critical point of I for + √ < p ≤ . Motivated by the above works [14,21,22,30,32,36], in the present article we mainly study the existence and multiplicity of positive solutions for Eq. (P µ,λ ) can not require conditions f changes sign in R and lim |x|→∞ f (x) = f∞ < . Furthermore, the existence of least energy positive solutions with negative energy and asymptotic behavior of positive solutions are also discussed. The main method of this paper is to consider minimization on two distinct components of the Nehari manifold corresponding to Eq. P µ,λ . The approach to Eq. P µ,λ has been inspired by the papers of [12,13,44]. They used the Nehari manifold and brering maps to study the bifurcation phenomena for a nonlinear elliptic problem on bounded domains or R N . Since Eq. P µ,λ is on R N , its variational setting is characterized by lack of compactness. To overcome this di culty we apply a simpli ed version of the steep well method of [11] and concentration compactness principle of [27]. Furthermore, the rst eigenvalue of problem −∆u + µg (x) u = λa (x) u in R N is less than λ (a Ω ) , which indicates that the original method at [12,13,21] cannot be directly applied, thus we provide an approximation estimate of eigenvalue to prove that the existence of positive solution for Eq. P µ,λ when < λ < λ (a Ω ) .
Next, we now consider what happens as λ → λ − (a Ω ) or µ → ∞. Let Then we have the following result.
The second result is to establish the existence of multiple positive solutions for Eq. (P µ,λ ) with λ > λ (a Ω ).
Finally, we investigate the nature of least energy positive solution u ( ) µ,λ as λ → λ + (a Ω ) and µ → ∞. As mentioned in the introduction a curve of positive solutions bifurcates to the right at λ (a Ω ) when B (ϕ ) < and µ su ciently large. The following theorem implies that u ( ) µ,λ will lie on this branch and the concentration of of the solutions for Eq. (P µ,λ ) with λ > λ (a Ω ) .
and obtain the same conclusions as all the previous theorems under the same hypotheses and in addition f is change sign in Ω. Since the proofs are similarly, and so we leave it to the reader to check. Some progress on the existence of positive solutions to Eq. C µ,λ , can be refer to [40,41].
The plan of the paper is as follows. In Section 2, some preliminary results are presented and we discuss the Nehari manifold and examine carefully the connection between the Nehari manifold and the brering maps.
In Section 4, we discuss the case when λ > λ (a Ω ) . In particular, we prove that Theorems 1.3, 1.4. Throughout this paper we denote a strong convergence by "→" and a weak convergence by " ".

Variational setting and Preliminaries
In this section, we give the variational setting for Eq. P µ,λ . Let be equipped with the inner product and norm For µ > , we also need the following inner product and norm It is clear that · ≤ · µ for µ ≥ and set Xµ = X, · µ . It follows from conditions (V ) and (V ) and the Hölder and Sobolev inequalities that we have Moreover, using conditions (V ) and (V ) , and the Hölder and Sobolev inequalities again, we have for any r ∈ , * , where, S the best constant for the embedding of D , (R N ) in L * (R N ). Moreover, if we assume that u ∈ L Np N+α R N , then by the Hardy-Littlewood-Sobolev inequality ( see [20,25,26]) to the function |u| p ∈ L N N+α R N , we obtain, in view of the Hölder inequality and ( . ) , We use the variational methods to nd positive solutions of Eq. P µ,λ . Associated with the Eq. P µ,λ , we consider the energy functional J µ,λ : Because the energy functional J µ,λ is not bounded below on X, it is useful to consider the functional on the Nehari manifold (see [29]) Note that N µ,λ contains every nonzero solution of Eq. P µ,λ . It is useful to understand N µ,λ in terms of the stationary points of mappings of the form hu(t) = J µ,λ (tu)(t > ). Such a map is known as the brering map. It was introduced by Drábek and Pohozaev [17], and further discussed by Brown and Zhang [12]. It is clear that, if u is a local minimizer of J µ,λ , then hu has a local minimum at t = . Moreover, tu ∈ N µ,λ if and only if h u (t) = for u ∈ X \ { }. Thus, points in N µ,λ correspond to stationary points of the maps hu and so it is natural to divide N µ,λ into three subsets N + µ,λ , N − µ,λ and N µ,λ corresponding to local minima, local maxima and points of in exion of brering maps. We have and Hence if we de ne Moreover, by ( . ), if A µ,λ (u) and B (u) have the same sign, then hu has exactly one turning point at and if A µ,λ (u) and B (u) have opposite signs, then hu has no turning points. Thus, if A µ,λ (u) , B (u) > , then hu(t) > for t small and positive but hu(t) → −∞ as t → ∞; also hu(t) has a unique (maximum) stationary hu is strictly increasing (resp. decreasing) for all t > . Thus, we have the following results.
The following Lemma shows that minimizers on N µ,λ are critical points for J µ,λ in X.
Proof. The proof of Lemma 2.2 is essentially same as that in Brown and Zhang [12, Theorem 2.3] (or see Binding et al. [5]), so we omit it here.
In order to prove main results, we will use a special case of the classical Brezis-Lieb lemma [8] for Riesz potentials.

Lemma 2.3. (Brezis-Lieb lemma for the Riesz potential [28, Lemma 2.4]). Let {un} be a bounded sequence in
We need the following result. Proof. Since vn ≤ vn µn ≤ c . We may assume that there exists v ∈ X such that Moreover, xn → ∞, and hence, B (xn , R ) ∩ x ∈ R N : g < c → . By the Hölder inequality, we have which a contradiction. Thus, vn → v in L r R N for all ≤ r < * . Moreover, by ( . ) and Lemma 2.3, This completes the proof. Next, we consider the following eigenvalue problem We can approach this problem by a direct method and attempt to obtain nontrivial solutions of problem ( . ) as relative minima of the functional on the unit sphere in B = u ∈ X : R N au dx = . Equivalently, we may seek to minimize a quotient as follows Then, by ( . ) , which indicates that λ ,µ (a) ≤ λ (a Ω ) for all µ > . Then we have the following results.
Proof. Let {un} ⊂ X with R N au n dx = be a minimizing sequence of ( . ) , that is R N |∇un| + µgu n dx → λ ,µ (a) as n → ∞.
We now show that there exists a minimizer on N − µ,λ which is a critical point of J µ,λ (u) and so a nontrivial solution of Eq. P µ,λ . First, we de ne Note that a restriction of J µ,λ on H (Ω), and c λ (Ω) independent of µ. Since < λ < λ (a Ω ) , similar to the argument of ( . ), we can conclude that J µ,λ | H (Ω) is bounded below on M µ,λ (Ω). Moreover, H (Ω) ⊂ Xµ for all µ > , one can see that Taking D > c λ (Ω). Then we have for all µ ≥ µ . Furthermore, we have the following results.
Next, we now consider what happens as λ → λ − (a Ω ) or µ → ∞. As might be expected from the introduction the sign of B (ϕ ) plays an important role. We conclude the following results by considering the case where B (ϕ ) > . Proof. We may assume without loss of generality that ϕ µ = . For < λ < λ (a Ω ) , we must have that Thus, This completes the proof.
Next, we are ready to prove Theorem 1.2: (i) Since λn → λ − (a Ω ) and µ (λn) → ∞ as n → ∞, we have µn → ∞ as n → ∞. Firstly, we show that {un} is bounded. Suppose on the contrary. Then we may assume without loss of generality that un µn → ∞ as n → ∞. Let vn = un un µn . Since Now, we show that If not, then we may assume that and so u = kϕ for some k. Since B (u ) = |k| p B (ϕ ) = and B (ϕ ) > , it follows that k = and u ≡ . Therefore, un → in X.
(ii) Here we follow the argument in [9] (or see [45]) to investigate the concentration for positive solutions of Eq. P µ,λ . For any sequence µn → ∞, let un := u µn ,λ be the positive solutions of Eq. P µn ,λ obtained in Theorem 1.1. By Lemma 3.1 there exists a positive constants c and C are independent of µn such that un µn ≤ c and J µn ,λ (un) ≥ C .Therefore, by Lemma 2.4, we may assume that there exists u ∈ H (Ω) such that un u in X and un → u in L r R N for all ≤ r < * . Now for any φ ∈ C ∞ (Ω), because J µn ,λ (un) , φ = , it is easy to check that On the other hand, the weakly lower semi-continuity of norm yields u ≤ lim inf n→∞ un ≤ lim n→∞ un µn , and thus, un → u in X. Moreover, by J µn ,λ (un) ≥ C > , one has u ≠ , which completes the proof.

The Proof of Theorems 1.3, 1.4 (λ > λ ( a Ω) )
If λ > λ (a Ω ), then Hence, if B (ϕ ) < , then by Lemma 2.1, N + µ,λ ≠ ∅. Thus, as well shall see, N µ,λ may consist of two distinct components in this case which makes it possible to prove the existence of at least two positive solutions by showing that J µ,λ has an appropriate minimizer on each component.
When N µ,λ = ∅, any non-zero minimizer for J µ,λ on N + µ,λ (or on N − µ,λ ) is also a local minimizer on N µ,λ and so will be a critical point for J µ,λ on N µ,λ and a solution of Eq. P µ,λ . We next show that, if N µ,λ = ∅, it is possible to obtain more information about the nature of the Nehari manifold. Since B (ϕ ) < , we can obtain that N + µ,λ ≠ ∅ for all µ > . Furthermore, we have the following results. Moreover, by Fatou's Lemma, which indicates that v ∈ A µ,λ . We now show that vn → v in Xµ . Suppose on the contrary. Then µ,λ is uniform bounded for µ > su ciently large.
Now we prove that un → u in Xµ . Let vn = un − u . Then vn in Xµ . By the Sobolev and Gagliardo-Nirenberg inequalities, for any µ > µ we have that where Πµ = C µc Since < N+α N < p < * α , it follows from ( . ) , ( . ) and ( . ) that Since wn µn → +∞, it follows that B (vn) → and so B (v ) = . We now show that vn → v in X. Suppose otherwise, then by ( . ) and ( . ) , Thus, v ≠ and for every µ > , there holds v ∈ A µ,λ ∩ B µ,λ , which is impossible. Hence vn → v in X. It follows that v µ = , Thus, for every µ > , there holds v ∈ A µ,λ ∩ B µ,λ which is impossible as A µ,λ ∩ B µ,λ = ∅. Hence, every minimizing sequence for J µ,λ (u) on N − µ,λ is bounded for µ su ciently large. (iii) Assume that inf u∈N − µ,λ J µ,λ (u) = . Then by the Ekeland variational principle [19], there exists a minimizing sequence {un} ⊂ N − µ,λ such that Next, we are ready to prove Theorem 1.4: (i) Since N + µn ,λn is uniformly bounded, then {un} is bounded, from Lemma 2.4, we may assume that there exists u ∈ H (Ω) such that un u in X, un → u in L r (R N ) for all ≤ r < * and B (un) → B (u ) . We also have We now show that un → u in X. Suppose on the contrary. Then