Multiple Solutions for Superlinear Fractional Problems via Theorems of Mixed Type

1 Introduction

In this paper, we focus our attention on the multiplicity of the following two fractional problems:

and

where s∈(0,1), N>2⁢s, λ∈ℝ and Ω⊂ℝN is a smooth bounded open set. Here (-ΔΩ)s is the spectral Laplacian which is given by

where {φk}k∈ℕ denotes the orthonormal basis of L2⁢(Ω) consisting of eigenfunctions of -Δ in Ω with homogeneous Dirichlet boundary conditions associated to the eigenvalues {αk}k∈ℕ, that is,

The fractional Laplacian operator (-ΔℝN)s may be defined for any u:ℝN→ℝ belonging to the Schwarz space 𝒮⁢(ℝN) of rapidly decaying C∞ functions in ℝN by

where CN,s is a normalizing constant depending only on N and s; see [15, 22] for more details.

As observed in [30], these two operators are completely different. Indeed, the spectral operator (-ΔΩ)s depends on the domain Ω considered, while the integral one (-ΔℝN)s evaluated at some point is independent on the domain in which the equation is set. Moreover, in contrast with the setting for the fractional Laplacian, it is not true that all functions are s-harmonic with respect to the spectral fractional Laplacian, up to a small error; see [1, 16] for more details.

Recently, many papers have appeared dealing with the existence and the multiplicity of solutions to problems driven by these two operators, by applying several variational and topological techniques. In particular, a great attention has been devoted to the study of fractional problems like (1.1) and (1.2) involving superlinear nonlinearities with subcritical or critical growth; see for instance [4, 5, 6, 3, 7, 8, 9, 10, 12, 22, 26, 28, 29, 31, 34]. It is worth observing that a typical assumption to study this class of problems is to require that the nonlinearity f verifies the well-known Ambrosetti–Rabinowitz condition [2], that is, there exist μ>2 and R>0 such that

This condition is quite natural and fundamental not only to guarantee that the Euler–Lagrange functional associated to the problem under consideration has a mountain pass geometry, but also to show that the Palais–Smale sequence of the Euler–Lagrange functional is bounded. We recall that (1.3) is somewhat restrictive and eliminates many nonlinearities. For instance the function

is superlinear at infinity but does not verify the condition (1.3).

The purpose of this paper is to investigate the multiplicity for the above two fractional problems when the parameter λ lies in a suitable neighborhood of any eigenvalue of the fractional operator under consideration, and f is superlinear and subcritical, but does not fulfill (1.3).

More precisely, along the paper we assume that f:Ω¯×ℝ→ℝ is a continuous function satisfying the following conditions:

1. there exist c1>0 and p∈(1,2s*-1), with 2s*=2⁢NN-2⁢s, such that

   | f ⁢ ( x , t ) | ≤ c 1 ⁢ ( 1 + | t | p )   for any  ⁢ ( x , t ) ∈ Ω × ℝ ;

2. lim | t | → 0 ⁡ f ⁢ ( x , t ) | t | = 0   uniformly in  ⁢ x ∈ Ω ;

3. lim | t | → ∞ ⁡ F ⁢ ( x , t ) t 2 = + ∞   uniformly in  ⁢ x ∈ Ω ,

   where

   F ⁢ ( x , t ) = ∫ 0 t f ⁢ ( x , τ ) ⁢ 𝑑 τ ;

4. there exist β∈(2⁢N⁢pN+2⁢s,2s*), c2>0 and T>0 such that

   f ⁢ ( x , t ) ⁢ t - 2 ⁢ F ⁢ ( x , t ) > 0   ⁢ for any  ⁢ x ∈ Ω , | t | > 0 ,

   f ⁢ ( x , t ) ⁢ t - 2 ⁢ F ⁢ ( x , t ) ≥ c 2 ⁢ | t | β   ⁢ for any  ⁢ x ∈ Ω , | t | ≥ T ;

5. F ⁢ ( x , t ) ≥ 0 for any (x,t)∈Ω×ℝ.

As a model for f we can take the function defined in (1.4).

Now we state our first main result regarding the multiplicity for problem (1.1):

In order to prove Theorem 1.1, we apply suitable variational methods after transforming problem (1.1) into a degenerate elliptic equation with a nonlinear Neumann boundary condition by using the extension technique [11, 12, 13, 14]. Thanks to this approach we are able to overcome the nonlocality of the operator (-ΔΩ)s and we can use some critical point results to study the extended problem. More precisely, we show that the functional associated to the extended problem respects the geometry required by the ∇-Theorem introduced by Marino and Saccon in [19]. Roughly speaking, this theorem says that if a C1-functional I defined on a Hilbert space has a linking structure and ∇⁡I verifies an appropriate condition on some suitable sets (see Definition 2.7 below), then I has two nontrivial critical points which may have the same critical level. We will apply this abstract result to the functional associated to the extended problem and we will get the existence of two nontrivial solutions. Finally, exploiting an additional linking structure, we get the existence of a third nontrivial solution. We recall that in the local setting, similar arguments have been developed and applied in many situations to obtain multiplicity results for several and different problems such as elliptic problems of second and fourth order, noncooperative elliptic systems, nonlinear Schrödinger equations with indefinite linear part in ℝN, variational inequalities; see [18, 20, 25, 32, 33]. Differently from the classic case, in the nonlocal framework, the only result comparable to Theorem 1.1 is due to Mugnai and Pagliardini [24] who obtained a multiplicity result to problem (1.1) when s=12 and f satisfies (1.3).

Our second main result concerns the multiplicity of solutions to (1.2).

The proof of the above result is obtained following the approach developed to prove Theorem 1.1. Anyway, we do not make use of any extension method and our techniques work also when we replace (-ΔℝN)s by the more general integro-differential operator -ℒK defined up to a positive constant as

where 𝒦:ℝN∖{0}→(0,∞) is a measurable function such that 𝒦⁢(-x)=𝒦⁢(x) for all x∈ℝN∖{0}, m⁢𝒦∈L1⁢(ℝN) with m⁢(x)=min⁡{|x|2,1}, and there exists θ>0 such that 𝒦⁢(x)≥θ⁢|x|-(N+2⁢s) for all x∈ℝN∖{0}.

In this context, we take care of the well-known results on the spectrum of integro-differential operators obtained by Servadei and Valdinoci in [29, 30]. We point out that in a recent paper Molica Bisci et al. [21] proved a similar result to Theorem 1.2 when f verifies condition (1.3), obtaining a nonlocal counterpart of the multiplicity result established in [23].

The paper is organized as follows. In Section 2, we recall some useful results related to the extension method in a bounded domain and then we provide some useful lemmas which will be fundamental to apply a critical point theorem of mixed nature. In Section 3, we deal with the existence of three nontrivial weak solutions to problem (1.2).

2 Multiplicity for Problem (1.1)