LEAST ENERGY SOLUTIONS OF NONLINEAR SCHRÖDINGER EQUATIONS INVOLVING THE HALF LAPLACIAN AND POTENTIAL WELLS

In this paper, we are concerned with the existence of least energy solutions of nonlinear Schrödinger equations involving the half Laplacian (−∆)u(x) + λV (x)u(x) = u(x)p−1, u(x) ≥ 0, x ∈ R , for sufficiently large λ, 2 < p < 2N N−1 for N ≥ 2. V (x) is a real continuous function on RN . Using variational methods we prove the existence of least energy solution u(x) which localize near the potential well int(V −1(0)) for λ large. Moreover, if the zero sets int(V −1(0)) of V (x) include more than one isolated components, then uλ(x) will be trapped around all the isolated components. However, in Laplacian case, when the parameter λ large, the corresponding least energy solution will be trapped around only one isolated component and become arbitrary small in other components of int(V −1(0)). This is the essential difference with the Laplacian problems since the operator (−∆)1/2 is nonlocal.

1. Introduction and main results.We are concerned with the following nonlinear Schrödinger equations involving the half Laplacian here 2 < p < 2 := 2N N −1 for N ≥ 2, V (x) is the potential, which is a real valued function on R N .In recent years, much attention has been devoted to the study of the fractional Laplacian.The fractional powers of the Laplacian, which are called fractional Laplacian and correspond to Lévy stable processes, appear in anomalous diffusion phenomena in physics, biology as well as other areas.They occur in flame propagation, chemical reaction in liquids, population dynamics.Lévy diffusion processes have discontinuous sample paths and heavy tails, while Brownian motion has continuous sample paths and exponential decaying tails.These processes have been applied to American options in mathematical finance for modeling the jump processes of the financial derivatives such as futures, forwards, options, and swaps, see [2] and references therein.Moreover, 2 MIAOMIAO NIU AND ZHONGWEI TANG they play important roles in the study of the quasi-geostrophic equations in geophysical fluid dynamics.
There are many results which are concerned with the problems involving the fractional Laplacian.Firstly, we refer the readers to the work by Caffarelli and Silvestre [6], in which a new formulation of the fractional Laplacian through Dirichlet-Neumann maps was introduced.By this formulation, they transferred the nonlocal problem to a local problem defined in a higher half space.After their pioneering work, there are many investigations to the fractional Laplacian problem by using variational methods.For example, using variational methods, Cabré and Tan [5] established the existence of positive solutions for fractional problems in a bounded domain with power-type nonlinearities.We also refer the work by Juan Dávila, Manuel del Pino and Juncheng Wei [9], where they considered the following fractional problem where 0 < s < 1, 1 < q < N +2s N −2s and V (x) is a sufficiently smooth potential with inf ε > 0 is a small parameter.Via a Lyapunov-Schmidt variational reduction, they proved the existence of multiple spike solutions which as ε small concentrate at separate places in the case of stable critical points and the existence of multiple spikes which as ε small concentrate at the same points.
For the following related fractional Schrödinger equation with 0 < s < 1 and V : R N → R is the potential function, there are also many investigations.
We firstly refer the reader to the most recent paper by Frank, Lenzmann and Silvestre [17], where the authors obtained the uniqueness results to fractional Laplacian problem related to (1.3).For other results, see also Bona and Li [4], Cheng [7], de Bouard and Saut [10], Dipierro, Palatucci and Valdinoci [13], Felmer et al [14], Frank and Lenzmann [16], Maris [20] and references therein.We also refer the readers to the paper by Jin, Li and Xiong [18] , where the authors considered the following fractional Laplacian equations with lower order terms where a, b ∈ C α (B 1 ) with 0 < α ∈ N and 2s + α is not an integer.They proved some priori estimates results for the solutions of the above equation (1.4), such as the local Schauder estimates for nonnegative solutions.We also refer the work by Tan and Xiong [23], where they established a Harnack inequality in the case of u ∈ C 2 (B 1 ) ∩ C (B 1 ).
The analogue problem to (1.1) for the Laplacian, for instance, the following problem has been investigated widely in the last decades.Much attention has been devoted to the study of the existence and uniqueness for one-bump or multi-bump bound states of (1.5).In [15], using a Lyapunov-Schmidt reduction, Floer and Weinstein established the existence of a standing wave solution of (1.5) when N = 1, p = 3 and V (x) is a bounded function which has a non-degenerate critical point for sufficiently small > 0. Moreover they showed that u concentrates near the given non-degenerate critical point of V when tends to 0. Their method and results were later generalized by Oh [21], [22] to the higher-dimensional case with 2 < p < 2N N −2 and the existence of multi-bump solutions concentrating near several non-degenerate critical points of V as tends to 0 was obtained.We also refer to Ambrosetti, Badiale and Cingolani [1], Cingolani and Nolasco [8], Pino and Felmer [11], [12] for the Laplacian proplems.
Related to the equation (1.5), the second author [24] considered the following equation with eletra-magnetic field and potential wells Under some proper conditions on a(x) and A(x), he proved the existence of least energy solutions to problem (1.6) which localize near the potential well int(a −1 (0)) for λ large.Similar investigation to equation (1.6) but there is no eletra-magnetic field, one can refer the work by Bartsch and Wang [3].Now we are ready to present our main assumptions, we assume that: where µ denotes the Lebesgue measure on R N .
Before stating our main results, we firstly give some notations and remarks.
To treat the nonlocal problem (1.1), we will study a corresponding extension problem in one more dimension, which allows us to investigate problem (1.1) by studying a local problem via classical nonlinear variational methods.
Let us denote the closure of the set of smooth functions compactly supported in R N +1 + , by ), with respect to the norm .
And we also introduce the fractional Sobolev space H 1/2 (R N ) which is a Banach space with the norm . Let Then E is the Hilbert space under the inner product and the norm induced by the inner product (., .) is Indeed, for every v(x, y) ∈ E, we denote v(x, 0) be the trace of v(x, y) on R N and take Then by the definition of E, we have We take Then E λ is the Hilbert space under the inner product and the norm induced by the inner product (., .)λ is We mention that the half Laplician in the whole space is a well studied operator.
Consider the operator T : u → −∂yv(•, 0).Since ∂yv is still a harmonic function, if we apply the operator twice, we obtain Thus the operator T that maps the Dirichlet-type data u to the Neumann-type data −∂yv(•, 0) is actually the half Laplacian.In this way we can study problem (1.1) by variational methods for a local problem.More precisely, we will study the following boundary value problem in a half space: where ν is the unit outer normal to R N × {0}.If v satisfies (1.8), then the trace u on R N × {0} of the function v will be a solution of problem (1.1).By studying (1.8), we establish the results for (1.1).
The energy functional associated with (1.8) is defined by where v + denotes the positive part of v for every function v, in other words, v + = max{v, 0}.We define the Nehari manifold be the infimum of J λ on the Nehari manifold M λ .
For λ large, the following problem is some kind of limit problem of (1.1).We shall prove that there exists a least energy solution of (1.1) converging for λ → ∞ to a least energy solution of (1.9).Similarly, to consider the problem (1.9), we will study the following mixed boundary value problem in a half space: where ν is the unit outer normal to Ω × {0}.If v satisfies (1.10), then the trace u on Ω × {0} of the function v will be a solution of (1.9).
To consider problem (1.10), we define a subspace E 0 of E as follows Similarly, by the definition of E 0 , we also have tr The energy functional associated with (1.10) is defined by Comparing with the Nehari manifold M λ , we define the Nehari manifold be the infimum of Φ on the Nehari manifold N .
Definition 1.1.We say that a function u λ (x) =v λ (x, 0) is a least energy solution of (1.1) if c λ is achieved by v λ ∈ M λ which is a critical point of J λ .Similarly we say that a function u(x) = v(x, 0) is a least energy solution of (1.9) if c(Ω) is achieved by v ∈ N which is a critical point of Φ.Now we give our main results and which are: ) and (V 2 ) hold.Then for λ large, (1.1) has a least energy solution u λ (x) = v λ (x, 0) .Furthermore, any sequence λn (λn → ∞ as n → ∞), {u λn (x)} has a subsequence such that u λn converges in H 1/2 (R N ) along the subsequence to a least energy solution u of (1.9).
As in the case of the least energy solution of (1.1), any solutions of (1.1) converges for λ → ∞ towards solutions of (1.9).More precisely, we have the following result.
Our paper is organized as follows: In section 2, we give a compactness result, Section 3 is devoted to the "limit" problem and Section 4 contains the proofs of the main results.
We will use the same C to denote various generic positive constants, and we will use o(1) to denote quantities that tend to 0 as λ( or n) → ∞.
2. Compactness result.The main result in this section is the following compactness result.To begin with, we firstly give the definition of (P S)c condition and (P S)c sequence.
Definition 2.1.Let X be a Banach space, ϕ ∈ C 1 (X, R) and c ∈ R. The functional ϕ satisfies the (P S)c condition if any sequence {un} ⊆ X such that has a convergent subsequence.
We call a sequence {un} ⊆ X is a (P S)c sequence of a functional ϕ if (2.1) is satisfied.Now we give the following compactness result.
The proof consists of a series of lemmas which occupy the rest of this section.
Proof.From the definition of E and E λ , to show the lemma, we only need to prove the following estimate Let us denote and for the above fixed small δ 0 , Thus for any function v ∈ E λ , we obtain that and where we have used the assumption (V 2 ) and the well known Sobolev inequality.The inequality , where C 0 depends only on N. Therefore, we indeed have proved (2.2) by adding up to the two inequalities (2.4) and (2.5) together.Thus the proof of the lemma is completed.
The following Lemma shows that 0 is an isolated critical point of J λ .
Proof.First we prove that any (P S)c-sequence must be bounded.In fact, for n large enough which proves (2.6).
On the other hand, there is a constant C > 0 independent of λ ≥ λ 0 ≥ 0 such that and {vn} is a (P S)c-sequence of J λ , then lim sup Hence, vn λ < σ 1 for n large, then by (2.7) 2p is as required.
Lemma 2.5.There exists δ 0 > 0 such that any (P S)c-sequence {vn} of J λ with λ ≥ 0 and c > 0 satisfies Proof.From the proof of Lemma 2.4 we know that {vn} is bounded and hence Lemma 2.6.Let C 1 be fixed.Then for any ε > 0 there exists Λε > 0 and Rε > 0 such that if {vn} is a where Proof.For R > 0, we set Then by Lemma 2.4, we can obtain that Using the Hölder inequality and (2.6), we obtain that for 1 < q < N/(N − 1), where C = C(N, q) is a positive constant and q is such that 1/q + 1/q = 1.µ(B(R)) denotes the Lebesgue measure of B(R).Setting θ := , the interpolation inequality and the Sobolev trace inequality yield From (2.10), the first summand on the right can be made arbitrarily small if λ large.On the other hand, from (2.11), the second summand on the right will be arbitrarily small if R large since µ(B(R)) → 0 as R → ∞ by assumption (V 2 ).This completes the proof.
The following lemma is well known and we only give the result without proof.
) For R > 0 and w ∈ E λ , we obtain from the Hölder's inequality, we have We also have that Thus, for every ε > 0, there exists R > 0 such that, for every It follows from the Rellich imbedding theorem, up to a subsequence, Then there exists a subsequence of vn (we still denote vn) and g ∈ L p (B R ) such that It is easy to obtain It follows from Lebesgue dominated convergence theorem that lim .
Lemma 2.9.Let λ ≥ λ 0 > 0 be fixed and let {vn} be a (P S)c-sequence of J λ .Then up to a subsequence, vn v in E λ with v being a weak solution of (1.8).Moreover, Proof.Firstly, by Lemma 2.4 we know that {vn} is bounded in E λ and hence {vn} is bounded in E.Then, up to a subsequence, vn v in E as n → ∞.We recall (1.7) and obtain ) where 2 = 2N N − 1 is the critical Sobolev exponent.Thus for any w ∈ E λ we have To show (2.22), we observe that Proof.Indeed, by Ekeland Variational Principle, there is P.S. sequence vn ∈ E 0 such that Φ(vn) → c(Ω) and Φ (vn) → 0.
Thus by Lemma 3.1, one can easily find a subsequence of {vn} ( we still denote vn) such that vn → v in E 0 and v achieves c(Ω).Then v is a least energy solution of (3.1).
Remark 1.When the zero set Ω = intV −1 (0) has more than one isolated components, for instance Ω = Ω 1 ∪ Ω 2 with Ω 1 ∩ Ω 2 = ∅.Then we have v(x, 0) 0 both in Ω 1 and in Ω 2 .Indeed, suppose on the contrary that v ∈ N is the least energy solution of (3.1) with v(x, 0) = 0 in Ω 1 and v(x, 0) 0 in Ω 2 .Then on one hand, (−∆) This contradiction shows that the least energy solution of (3.1) satisfies v(x, 0) 0 both in Ω 1 and in Ω 2 .The phenomenon is an essential difference from the local operator Laplacian since in Laplacian case, u = 0 in Ω immediately indicates that ∆u = 0 in Ω for any domain Ω.
Remark 2. By Rermark 1 combining Hopf's Lemma, One can easily check that u(x) = v(x, 0) > 0 for all x ∈ Ω. 4. Proofs of main results.In this section we will give the proofs of our main results.To begin with, we firstly give an asymptotic behavior for c λ as λ large.More precisely, we have the following lemma: Proof.It is easy to see that c λ ≤ c(Ω) for all λ ≥ 0. It is not difficult to check that c λ is monotone increasing with respect to λ > 0 according to the definition of c λ : First of all, Lemma 2.4 implies k > 0 and by Corollary 1, for n large enough, there exists a sequence vn ∈ M λn which is a solution of (1.8) with λ being replaced by λn such that J λn (vn) = c λn .Similar to the proof of Lemma 2.4, it is easy to verify that {vn} is bounded in E, thus we may assume that vn v in E and vn(x, 0) → v(x, 0) in L q loc (R N ) for 2 ≤ q < 2 .However, since V (x) ≥ ε 0 > 0 for all x ∈ F and for some ε 0 > 0, it follows that