Solutions for Fractional Schrödinger Equation Involving Critical Exponent via Local Pohozaev Identities

1 Introduction

In this paper, we are concerned with the following problem:

where N≥3 and 2s*=2⁢NN-2⁢s is the critical Sobolev exponent. For any s∈(0,1), (-Δ)s is the fractional Laplacian in ℝN, which is a nonlocal operator defined as

where P.V. denotes the Cauchy principal value and c⁢(N,s) is a constant depending on N and s. This operator is well defined in Cloc1,1∩ℒs, where ℒs={u∈Lloc1:∫ℝN|u⁢(x)|1+|x|N+2⁢s⁢𝑑x<∞}. For more details on the fractional Laplacian, we referee to [16, 22] and the references therein.

The fractional Laplacian operator appears in divers areas, including biological modeling, physics and mathematical finances, and can be regarded as the infinitesimal generator of a stable Lévy process (see, for example, [5]). From the view point of mathematics, an important feature of the fractional Laplacian operator is its nonlocal property, which makes it more challenging than the classical Laplacian operator. Thus, problems with the fractional Laplacian have been extensively studied, both for the pure mathematical research and in view of concrete real world applications, see, for example, [1, 2, 4, 3, 11, 10, 9, 8, 7, 17, 19, 29, 23, 37, 39, 40, 44] and the references therein.

Solutions of (1.1) are related to the existence of standing wave solutions to the following fractional Schrödinger equation:

That is, solutions with the form Ψ⁢(x,t)=e-i⁢c⁢t⁢u⁢(x), where c is a constant.

In this paper, under a weaker symmetry condition for V⁢(y), we will construct multi-bump solutions for (1.1) through a finite-dimensional reduction method, combined with various local Pohozaev identities. More precisely, we consider V⁢(y)=V⁢(|y′|,y′′)=V⁢(r,y′′), y=(y′,y′′)∈ℝ2×ℝN-2 and assume that:

1. V ⁢ ( y ) ≥ 0 is a bounded function that belongs to C2⁢(ℝN), and r2⁢s⁢V⁢(r,y′′) has a critical point (r0,y0′′) satisfying r0>0, V⁢(r0,y0′′)>0 and deg⁢(∇⁡(r2⁢s⁢V⁢(r,y′′)),(r0,y0′′))≠0.

Since the fractional operator is nonlocal, we have to overcome more difficulties than the Laplace equation. Such as, we need to study the corresponding harmonic extension problem and deal with (-Δ)s⁢(ϕ⁢φ) in an appropriate process. Hence, some new ideas and techniques are needed. We will explain these later.

Before state the main results, let us first introduce some notations. Denote Ds⁢(ℝN) the completion of C0∞⁢(ℝN) under the norm ∥(-Δ)s2⁢u∥L2⁢(ℝN), where ∥(-Δ)s2⁢u∥L2⁢(ℝN) is defined by (∫ℝN|ξ|2⁢s|𝒢u(ξ)|2dξ)12, and 𝒢⁢u is the Fourier transformation of u:

We will construct the solutions in the following the energy space:

with the norm

We define the functional I on Hs⁢(ℝN) by

where (u)+=max⁡(u,0). Then the solutions of problem (1.1) correspond to the critical points of the functional I.

It is well known that the functions

where C⁢(N,s)=2N-2⁢s2⁢Γ⁢(N+2⁢s2)/Γ⁢(N-2⁢s2), are the only solutions of the problem (see [32])

Define

Let

To construct the solution of (1.1), we hope to use Ux,λ⁢(y) as an approximation solution. However, the decay of Uxj,λ is not fast enough for us when N≤6⁢s. So, we need to cut off this function. Let δ>0 be a small constant such that r2⁢s⁢V⁢(r,y′′)>0 if |(r,y′′)-(r0,y0′′)|≤10⁢δ. Let ζ⁢(y)=ζ⁢(r,y′′) be a smooth function satisfying ζ=1 if |(r,y′′)-(r0,y0′′)|≤δ, ζ=0 if |(r,y′′)-(r0,y0′′)|≥2⁢δ, |∇⁡ζ|≤C and 0≤ζ≤1. Denote

Let

Then a direct computation shows that

We obtain the following result when N>4⁢s.

We are also interested in the case N=4⁢s. Since s<1, we have N=3=4⁢s. This case is corresponding to the Laplace equation with N=4.

When N>4⁢s, we introduce the following norms:

and

where τ=N-4⁢s2⁢(N-2⁢s). When N=3=4⁢s, we use the following norms:

and

where N-2⁢s2=34 and N+2⁢s2=94.

We will prove Theorems 1.1 and 1.3 by a finite-dimensional reduction method, combined with various local Pohozaev identities. The finite-dimensional reduction method has been extensively used to construct solutions for equations with critical growth. We refer to [6, 15, 14, 13, 12, 18, 21, 24, 25, 26, 28, 30, 31, 33, 34, 36, 43, 42, 41] and the references therein. Roughly speaking, the outline to carry out the reduction argument is as follows: We first construct a good enough approximation solution and linearize the original problem around the approximation solution. Then we solve the corresponding finite-dimensional problem to obtain a true solution.

To finish the second step, we have to obtain some good enough estimates in the first step. In our case, since the fractional operator is nonlocal, we have to overcome more difficulties than the Laplace equation. One of them is that we will use Zr¯,y¯′′,λ as an approximate solution, but we have to deal with (-Δ)s⁢(ζ⁢(y)⁢Uxj,λ⁢(y)) and (-Δ)s⁢(ζ⁢(y)⁢Zj,t⁢(y)). By (1.2), we can deduce that

In order to obtain a good enough estimate with ∥⋅∥**, we need to deal with some concrete difficulties, and devote ourselves to calculate the last principal value very carefully (see Lemma 2.5). More precisely, when N>4⁢s, we need to show that

Before the end of this introduction, we briefly outline the proof for the case of N>4⁢s and point out some other difficulties (the idea of the proof for the other case is similar but with different estimates). We first use Zr¯,y¯′′,λ as an approximate solution to obtain a unique function ϕ⁢(r¯,y¯′′,λ). Then the problem of finding critical points for I⁢(u) can be reduced to that of finding critical points of F⁢(r¯,y¯′′,λ)=I⁢(Zr¯,y¯′′,λ+ϕ⁢(r¯,y¯′′,λ)). In the second step, we solve the corresponding finite-dimensional problem to obtain a solution. However, we can only obtain ∥ϕ∥*≤Cλs+σ (see Proposition 2.3). From Lemmas B.3 and B.5, we know that

and

Note that the estimate of ϕ is only good enough for the expansion (1.3), but it destroys the main terms in the expansions of (1.4) and (1.5). To overcome this difficulty, following the idea in [35], instead of studying (1.4) and (1.5), we turn to study the following local Pohozaev identities:

and

where uk=Zr¯,y¯′′,λ+ϕ, u~k is the extension of uk (see below (1.8)),

For any u∈Ds⁢(ℝN), u~ is defined by

where

with constant β⁢(N,s) such that ∫ℝNPs⁢(x,1)⁢𝑑x=1. Moreover, u~ satisfies (see [11])

and

where ωs=21-2⁢s⁢Γ⁢(1-s)/Γ⁢(s).

Due to the nonlocality of the fractional Laplacian operator, we can not built a local Pohozaev identity for problem (1.1). So, we need to study the corresponding harmonic extension problem (1.9) and (1.10). The relationship between u and u~ is (1.8). Hence, we have to give some estimates for this kind of integrals. The local Pohozaev identities (1.6) and (1.7) are much more complicated. We have to integrate one more time than the Laplacian operator case. This is very difficult when we derive some sharp estimates for each term in (1.6) and (1.7). We need a lot of preliminary lemmas. For example, some suitable estimates on ∇⁡Z~xi,λ and ∇⁡ϕ~ are established in Lemmas A.5 and A.6.

Our paper is organized as follows. In Section 2, we perform a finite-dimensional reduction. We prove Theorem 1.1 in Section 3. Theorem 1.3 is proved in Section 4. In Appendix A, we give some essential estimates. We put the energy expansions for 〈I′⁢(Zr¯,y¯′′,λ+ϕ⁢(r¯,y¯′′,λ)),∂⁡Zr¯,y¯′′,λ∂⁡λ〉, 〈I′⁢(Zr¯,y¯′′,λ+ϕ⁢(r¯,y¯′′,λ)),∂⁡Zr¯,y¯′′,λ∂⁡r¯〉 and 〈I′⁢(Zr¯,y¯′′,λ+ϕ⁢(r¯,y¯′′,λ)),∂⁡Zr¯,y¯′′,λ∂⁡y¯′′〉 in Appendix B.

2 Finite-Dimensional Reduction

In this section, we perform a finite-dimensional reduction by using Zr¯,y¯′′,λ as an approximation solution. We consider the following linearized problem:

for some numbers cl.