Multiplicity result for non-homogeneous fractional Schrodinger--Kirchhoff-type equations in ℝⁿ

1 Introduction

The aim of this article is to study a Schrödinger–Kirchhoff-type equation with fractional p-Laplacian in ℝn,

where 0<s<1<p<∞, p⁢s<n, and the operator (-Δ)ps is the fractional p-Laplacian which may be defined along a function φ∈C0∞⁢(ℝn) as

where x∈ℝn, and B⁢(x,ϵ)={y∈ℝn:|x-y|<ϵ}. We invite the reader to check [16, 19, 20, 21, 25, 35] and the references therein for further details on the fractional p-Laplace operator. The function g=g⁢(x) can be viewed as a perturbation term.

When p=2 and M=1, equation (1.1) becomes the fractional Laplacian equation in ℝn,

which has been study by many researchers, see, for instance, [2, 6, 7, 8, 11, 34] and the references therein for some results.

In recent years, great attention has been paid on the study of problems involving the non-local fractional Laplacian or more general integro-differential operators. This type of operators arises in a quite natural way in many different applications, such as continuum mechanics, phase transition phenomena, population dynamics, minimal surfaces and game theory, as they are the typical outcome of stochastical stabilization of Lévy processes, see [3, 4, 5, 23] and the references therein. The literature on fractional operators and their applications is very extensive, see, for example, [1, 27, 28, 2, 6, 7, 8, 15, 14, 11, 13, 12, 24, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40]. For a short introduction to the fractional Laplacian and the fractional Sobolev spaces, the reader is referred to [10]. Furthermore, research has been done in recent years for the regional fractional Laplacian, where the scope of the operator is restricted to a variable region near each point. We mention the work by Guan [17] and Guan and Ma [18] where they study these operators, their relation with stochastic processes and they develop an integration by parts formula. We also mention the work by Ishii and Nakamura [21], where they studied the Dirichlet problem for regional fractional Laplacian modeled on the p-Laplacian and the recent work by Felmer and Torres [13, 12], where they considered the existence, symmetry properties and concentration phenomena of solutions of the non-linear Schrödinger equation with non-local regional diffusion. These regional operators present various interesting characteristics that make them very attractive from the point of view of mathematical theory of non-local operators.

On the other hand, Fiscella and Valdinoci in [15] first proposed a stationary Kirchhoff variational equation which models the non-local aspect of the tension arising from non-local measurements of the fractional length of the string. Indeed, problem (1.1) is a fractional version of a model, the so-called Kirchhoff equation, introduced by Kirchhoff in [22]. More precisely, Kirchhoff established a model given by the problem

where ρ is the mass density, p0 is the initial tension, h is the area of the cross section, E is the Young modulus of the material and L is the length of the string, which extends the classical D’Alembert wave equation by considering the effects of the changes in the length of the strings during the vibrations. Note that non-local boundary problems like (1.2) can be used to model several physical and biological systems where u describes a process, which depend on the average of itself, such as the population density [9]. A parabolic version of problem (1.2) can be used to describe the growth and movement of a particular species. The movement, modeled by the integral term, is assumed to be dependent on the energy of the entire system with u being its population density. Alternatively, the movement of a particular species may be subject to the total population density within the domain (for instance, the spreading of bacteria) which gives rise to equations of the type

It is worth pointing out that problem (1.2) received much attention only after Lions [26] proposed an abstract framework to it. For some motivation in the physical background for the fractional Kirchhoff model, we refer to [15, Appendix A]. Also, see for example [29, 30, 31, 37] for non-degenerate Kirchhoff-type problems, and [1, 36], for degenerate Kirchhoff-type problems in this direction.

Very recently in [32], Pucci, Xiang and Zhang considered problem (1.1) under the following assumptions: For the Kirchhoff function M, suppose that:

1. M∈C⁢(ℝ0+) satisfies inft∈ℝ0+⁡M⁢(t)≥a>0, where a>0 is a constant.

2. There exists θ∈[1,nn-s⁢p) such that

   θ⁢ℳ⁢(t)=θ⁢∫0tM⁢(s)⁢𝑑s≥t⁢M⁢(t) for all ⁢t∈ℝ0+.

For the potential V, suppose that:

1. V∈C⁢(ℝn,ℝ) and infx∈ℝn⁡V⁢(x)≥V0>0, where V0>0 is a constant.

2. There exists h>0 such that

   lim|y|→∞⁡meas⁢({x∈Bh⁢(y):V⁢(x)≤c})=0 for any ⁢c>0.

For the nonlinearity f, suppose that:

1. f:ℝn×ℝ→ℝ is a Carathéodory function and there exist q, with θ⁢p<q<ps*, and a1>0 such that |f⁢(x,t)|≤a1⁢(1+|t|q-1) for a.e. x∈ℝn and all t∈ℝ.

2. There exists μ>θ⁢p such that

   μ⁢F⁢(x,t)≤t⁢f⁢(x,t) for all x∈ℝn and all t∈ℝ.

3. f⁢(x,t)=o⁢(|t|p-1) as t→0, uniformly for x∈ℝn.

4. infx∈ℝn,|t|=1⁡F⁢(x,t)>0.

Under the assumptions (M1)–(M2), (V1)–(V2), (f1)–(f4), the authors first establish a Batsch–Wang-type compact embedding theorem for the fractional Sobolev spaces. Multiplicity results are then obtained by using the Ekeland’s variational principle and the Mountain Pass Theorem. Also we mention the work by Xiang, Zhang and Ferrara [38], where they considered the non-homogeneous fractional p-Kirchhoff equations with concave-convex nonlinearities and have obtained the existence of two non-trivial entire solutions by applying the mountain pass theorem and Ekeland’s variational principle.

Inspired by these previous works, in this paper we consider problem (1.1) under some weaker assumptions. More precisely, we assume that the Kirchhoff function M satisfies (M1)–(M2). Furthermore, regarding the potential V we only assume (V1) and for the functions f we suppose that:

1. f∈C⁢(ℝn×ℝ,ℝ) and there exists ps*>μ>θ⁢p such that

   0<μ⁢F⁢(x,ζ)≤ζ⁢f⁢(x,ζ),for all x∈ℝn and all ζ∈ℝ∖{0},

   where F⁢(x,ζ)=∫0ζf⁢(x,s)⁢𝑑s.

2. There exists some positive continuous function a:ℝn→ℝ with

   (1.3)lim|x|→∞⁡a⁢(x)=0

   such that

   |f⁢(x,ζ)|≤a⁢(x)⁢|ζ|μ-1 for all⁢(x,ζ)∈ℝn×ℝ.

Before stating our main results, we introduce some useful notations. First of all, define the Gagliardo seminorm by

where u:ℝn→ℝ is a measurable function. Now, let the fractional Sobolev space be denoted as

and assume that it is endowed with the norm

where the fractional critical exponent is defined by

Moreover, we consider the fractional Sobolev space with potential

endowed with the norm

We say that u∈Xs is a weak solution of problem (1.1) if

for any φ∈Xs.

From here on we set p′=p/(p-1), the Hölder conjugate of p. Our main result in this paper is the following theorem.

The remaining part of this paper is organized as follows. Some preliminary results are presented in Section 2. In Section3, we are devoted to accomplishing the proof of Theorem 1.1.

2 Preliminaries

To study the fractional problem (1.1), the so-called fractional Sobolev spaces Ws,p⁢(ℝn) with 0<s<1 are expedient. If 1<p<∞, as usual, the norm is defined through

We recall the Sobolev embedding theorem.

Consider now the space Xs defined by

endowed with the norm

Now we introduce some more notations and necessary definitions. Let ℬ be a real Banach space. If I is a continuously Fréchet-differentiable functional defined on ℬ, we write I∈C1⁢(ℬ,ℝ). Recall that I∈C1⁢(ℬ,ℝ) is said to satisfy the (PS) condition if every sequence {un}n∈ℕ⊂ℬ for which {I⁢(un)}n∈ℕ is bounded and I′⁢(un)→0 as n→∞ possesses a convergent subsequence in ℬ.

Moreover, let Br be the open ball in ℬ with radius r and centered at 0, and let ∂⁡Br denote its boundary. Under the conditions of Theorem 1.1, we obtain the existence of the first weak solution of (1.1) using the following well-known Mountain Pass Theorem, see [33].

As far as the second solution is concerned, we obtain it using the minimizing method, which is contained in a small ball centered at 0.

3 Proof of Theorem 1.1

The aim of this section is to establish the proof of Theorem 1.1. For this purpose, we are going to establish the corresponding variational framework to obtain solutions of (1.1). To this end, define the functional I:Xs→ℝ by

where

If (M1) and (V1) hold, then J:Xs→ℝ is of class C1⁢(Xs) and

(3.2)

for all u,v∈Xs. Moreover, J is weakly lower semi-continuous in Xs, see [32].

Now we prove our key lemma.

To prove Theorem 1.1 we first consider some lemmas.