Ground states for fractional Schrödinger equations involving a critical nonlinearity

1 Introduction and main result

In this paper, we study ground state solutions for the following fractional Schrödinger equations with a critical nonlinearity:

where 2α*=2⁢NN-2⁢α with N≥2, λ>0, α∈(0,1) and (-Δ)α is the fractional Laplacian operator, which (up to normalization constants) may be defined as

where P.V. stands for the principal value.

The fractional Laplacian operator (-Δ)α can be seen as the infinitesimal generators of Lévy stable diffusion processes (see [1]). The Lévy processes occur widely in physics, chemistry, biology and finance, see for example [11, 22]. Some interesting topics concerning the fractional Laplacian such as the nonlinear fractional Schrödinger equation (see [17, 18, 22, 23, 26, 51]), the nonlinear fractional Kirchhoff equation (see [20, 29, 34, 35, 48, 49, 50]), the fractional porous medium equation (see [13, 46]) and so on, have attracted recently much research interest. Indeed, the literature on fractional operators and their applications to partial differential equations is quite large. Here we would like to mention a few, see for instance [4, 9, 15, 26, 27, 30, 31, 36] for recent results.

In the celebrated paper [8], Berestycki and Lions studied the following classical nonlinear scalar field equation:

where N≥3. Using certain assumptions on g, which are now named Berestycki–Lions conditions, they proved the existence of a ground state solution. Using Pohozaev identity, they also showed that Berestycki–Lions conditions are almost necessary for the existence of a solution for problem (1.2). For N=2, Berestycki, Gallouët and Kavian [7] obtained the existence of a radially symmetric positive solution of (1.2) under some appreciate conditions on g. In fact, the authors in [8, 7] just dealt with the subcritical case. However, for the critical case, the problem becomes very difficult due to the loss of the compactness of the embedding H1⁢(ℝN)↪L2*⁢(ℝN). About the characterization of ground state solutions corresponding to the Berestycki–Lions (and others) result for the critical growth case, for example, we refer to [2, 52].

In [14], using minimax arguments, Chang and Wang study the following scalar field equation involving the fractional Laplacian:

They obtained a positive ground state under the fractional version of Berestycki–Lions type assumptions, in which g is subcritical at infinity. On some recent works involving the subcritical case, we refer to, for instance, [17, 37] and references therein.

In [42], Shang and Zhang studied the existence and multiplicity of solutions for the critical fractional Schrödinger equation:

Based on variational methods, they showed that problem (1.4) has a nonnegative ground state solution for all sufficiently large λ and small ε. In this paper, the following monotone condition was imposed on the continuous subcritical nonlinearity f:

In [44], Shen and Gao obtained the existence of nontrivial solutions for problem (1.4) under various assumptions on f⁢(t) and the potential function V⁢(x). Indeed, the authors assumed the well-known Ambrosetti–Rabinowitz condition ((AR) condition for short) on f:

where F⁢(t)=∫0tf⁢(s)⁢𝑑s. See also the recent papers [37, 38] on the fractional Schrödinger equations with or without (AR) condition. In [45], Teng was concerned with the following fractional Schrödinger equations involving a critical nonlinearity:

where 2<p<2α*, and the potential functions A⁢(x) and B⁢(x) satisfy certain hypotheses. Using the s-harmonic extension technique of Caffarelli and Silvestre [12], the concentration-compactness principle of Lions [24] and methods of Brézis and Nirenberg [10], the author obtained the existence of ground state solutions. On fractional Kirchhoff problems involving critical nonlinearity, see for example [3, 33] for some recent results. Last but not least, fractional elliptic problems with critical growth, in a bounded domain, have been studied by some authors in the last years, see [5, 6, 21, 43, 41, 40] and references therein.

On the other hand, Feng in [19] investigated the following fractional Schrödinger equations:

where 2<p<2α*, V⁢(x) is a positive continuous function. By using the fractional version of the concentration-compactness principle of Lions [24], the author obtained the existence of ground state solutions to problem (1.8) for some λ>0.

Motivated by the above works, we are interested in the existence of ground state solutions for problem (1.1) via concentration compactness principle in the fractional Sobolev space (see [32, Theorem 1.5]), which is another fractional version of Lions [25]. To this end, we impose the following conditions on f:

1. f∈C⁢(ℝ,ℝ) and for any t≤0, f⁢(t)=0.

2. limt→0+⁡f⁢(t)t=0 and limt→+∞⁡f⁢(t)t2α*-1=0.

3. There exists q>2 such that for any t≥0, f⁢(t)≥tq-1.

4. For any t>0, 0<2⁢F⁢(t)≤f⁢(t)⁢t.

In order to obtain a nonnegative solution, we assume that f⁢(t)=0 for any t≤0 throughout the paper. From (H2) we know that f is subcritical. Moreover, the solution to problem (1.1) is obtained without assuming the classical condition (1.5) or (1.6). Clearly, we employ the weaker condition (H4) on f to replace (AR) condition. A typical example for f is given by f⁢(t)=tq-1 for t≥0 with q>2.

Now, we give the definition of weak solutions for problem (1.1).

We define the following functionals on Hα⁢(ℝN):

and

It is easy to check that I∈C1⁢(Hα⁢(ℝN),ℝ) and the weak solution for problem (1.1) coincides with the critical point of I.

Note that I is neither bounded from above nor from below on Hα⁢(ℝN), it is difficult to look directly for critical points of I. In this paper, we will first consider a constrained minimization problem and obtain its minimizer in the fractional radially symmetric function space Hrα⁢(ℝN). Then, we verify that the minimizer under a scale change is a ground state solution for problem (1.1). Now we are ready to give our main result as follows.

This paper is organized as follows. In Section 2, we will give some necessary definitions and properties of fractional Sobolev spaces. In Section 3, by using the concentration-compactness principle and radially decreasing rearrangements, we give the proof of Theorem 1.3.

2 Preliminaries

For the convenience of the reader, in this part we recall some definitions and basic properties of fractional Sobolev spaces Hα⁢(ℝN). For a deeper treatment on these spaces and their applications to fractional Laplacian problems of elliptic type, we refer to [16, 28] and references therein.

We consider the Schwartz space 𝒮 of rapidly decaying C∞ functions in ℝN, with the corresponding topology generated by the seminorms

where φ∈𝒮⁢(ℝN). Let 𝒮′⁢(ℝN) be the set of all tempered distributions, that is the topological dual of 𝒮⁢(ℝN). As usual, for any φ∈𝒮⁢(ℝN), we denote by

the Fourier transform of φ and we recall that one can extend ℱ from 𝒮⁢(ℝN) to 𝒮′⁢(ℝN).

For any α∈(0,1), the fractional Sobolev space Hα⁢(ℝN) is defined by

endowed with the norm

where the term

is the so-called Gagliardo seminorm of u. The space H˙α⁢(ℝN) is defined as the completion of C0∞⁢(ℝN) under the norm [u]Hα⁢(ℝN).

Using the Fourier transform, the fractional Laplacian (-Δ)α can also be seen as a pseudo-differential operator of |ξ|α:

where ϕ∈𝒮.

Evidently, (2.1) means that the fractional Laplacian is nonlocal, which is a distinguished feature, and hence makes it difficult to deal with. It is worth mentioning that in a bounded domain, the Fourier definition of the fractional laplacian does not agree with its local Caffarelli–Silvestre interpretation (see [12]), we refer to [39] for a detailed discussion.

3 Proof of Theorem 1.3

Throughout this section, we assume that conditions (H1)–(H4) are satisfied. In this part, rather than looking for critical points of I, we will first consider the following constrained minimization problem.

We define

and

where Hrα⁢(ℝN)={u∈Hα⁢(ℝN):u⁢(x)=u⁢(|x|)} is the fractional radially symmetric function space and

The main difficulties here are that the embedding Hα⁢(ℝN)↪L2α*⁢(ℝN) is not compact and we do not have a similar radial lemma (see [8]) in Hrα⁢(ℝN). To get the compactness of bounded minimizing sequence in Hα⁢(ℝN), we assume that λ in (1.1) is large. Based on the concentration-compactness principle in Hα⁢(ℝN) (see [32]) and radially decreasing rearrangements in [17], we obtain a nonnegative radially symmetric minimizer for (3.1).

Next, using Pohozaev identity for (1.1) we will give a characterization of A.

In [14], using the α-harmonic extension, the authors proved the Pohozaev identity for (1.3) with subcritical nonlinearities. In this paper, although the problem (1.1) involves critical nonlinearities, similarly to the proof of Pohozaev identity in [14], we could also obtain the following Pohozaev identity for (1.1): Let u∈Hα⁢(ℝN) be a weak solution of (1.1), then

We introduce the set 𝒫 of nontrivial functions satisfying the Pohozaev identity (3.8), i.e.,

and

Set

and

Note that the embedding

is compact (see [14]). Hence, there exists 0≤ψ∈Hrα⁢(ℝN) such that ∥ψ∥Hα⁢(ℝN)=1 and ∥ψ∥Lq⁢(ℝN)=Cq-1, where q∈(2,2α*) and Cq is the best Sobolev constant of the above embedding, i.e.,

In the following, we will take a special minimizing sequence for A and get its compactness. Then, we could obtain a minimizer for (3.1).

From Ekeland’s variational principle (see [47, Theorem 8.5]), there exist {un}⊂ℳ and {λn}⊂ℝ such that