On fractional p-Laplacian problems with local conditions

1 Introduction and main results

In this paper we are concerned with the existence and multiplicity of nontrivial solutions of the following fractional p-Laplacian equation:

where λ∈(0,+∞), 0<s<1<p<+∞, Ω⊂ℝN is a bounded domain with smooth boundary, N⩾2, and f⁢(x,t) is a Carathéodory function defined on Ω×(-δ,δ), with δ>0 being small.

When p=2, much attention has been paid to the semi-linear problem

from the point of view of existence, non-existence and regularity, where g:Ω×ℝ↦ℝ is a Carathéodory function satisfying suitable growth conditions. Several existence results via variational methods are proved in a series of papers [18, 20, 19, 21]. The issues of regularity and non-existence of solutions are studied in [4, 15, 16, 14, 3]. The corresponding equations in ℝN have also been widely studied, see, for example [17, 5, 9, 1, 10] and references therein.

Very recently, a new nonlocal and nonlinear operator was considered, namely, for p∈(1,+∞), s∈(0,1) and u smooth enough,

which is consistent, up to some normalization constant depending upon N and s, with the linear fractional Laplacian (-Δ)s in the case p=2. This operator, known as the fractional p-Laplacian, leads naturally to the quasilinear problem (P1). One typical feature of this operator is the nonlocality, in the sense that the value of (-Δ)ps⁢u at any point x∈Ω depends not only on the values of u in Ω, but actually on the whole ℝN, since u⁢(x) represents the expected value of a random variable ties to a process randomly jumping arbitrarily far from the point x. While in the classical case, by the continuity properties of the Brownian motion, at the exit time from Ω one necessarily is on ∂⁡Ω, due to the jumping nature of the process, at the exit time one could end up anywhere outside Ω. In this sense, the natural nonhomogeneous Dirichlet boundary condition consists in assigning the values of u in ℝN∖Ω rather than merely on ∂⁡Ω. Then, it is reasonable to search for solution in the space of functions u∈Ws,p⁢(ℝN) vanishing outside Ω. It should be pointed out that in a bounded domain, this is not the only possible way to provide a formulation for the problem. In the works of [11, 13], the eigenvalue problem associated with (-Δ)ps is studied, and particularly some properties of the first eigenvalue and of the higher order (variational) eigenvalues are obtained. From the point of view of regularity theory, some results can be found in [13]. This work is most focused on the case where p is large and the solutions inherit some regularity directly from the functional embedding theorems. In [7, 2], relevant results about the local boundedness and Hölder continuity for the solutions to the problem of finding (s,p)-harmonic functions u were obtained. Very recently, in [12], Iannizzotto et al. established a priori L∞-bounds for the solutions of problem (P1) under suitable growth conditions on the nonlinearity.

In all the works mentioned above, the nonlinearity is assumed on the whole Ω×ℝ. Motivated by [12, 22, 6], we can consider problem (Pλ) with local conditions on the nonlinearity f. Firstly, we assume that f satisfies a p-sublinear condition at the origin, without any growth condition at infinity. In particular, we assume the following:

1. f⁢(x,t) is a Carathéodory function defined on Ω×(-δ,δ).

2. There exists positive constant p1∈(p2ps*,p) such that

   lim|t|→0⁡F⁢(x,t)|t|p1=+∞ uniformly for a.e. ⁢x∈Ω,

   where F⁢(x,t)=∫0tf⁢(x,ξ)⁢𝑑ξ.

3. There exists positive constant p2∈(p2ps*,p) such that

   lim|t|→0⁡F⁢(x,t)|t|p2=0 uniformly for a.e. ⁢x∈Ω.

4. There exists α∈(p1,p) such that

   α⁢F⁢(x,t)-f⁢(x,t)⁢t⩾0 for a.e. ⁢x∈Ω, with ⁢|t|⁢small.

The first two results of our paper read as follows.

Furthermore, if f⁢(x,t) is also odd in t, then for every λ∈(0,+∞), problem (Pλ) has infinitely many nontrivial weak solutions.

Secondly, we consider (Pλ) with p-superlinear nonlinearity. In particular, we make the following assumptions on f just near origin:

1. There exists q1∈(p,2⁢p⁢ps*p+ps*) such that

   lim|t|→0⁡f⁢(x,t)⁢t|t|q1=0 uniformly for a.e. ⁢x∈Ω.

2. There exists q2∈(p,p+γ) such that

   lim|t|→0⁡F⁢(x,t)|t|q2=+∞ uniformly for a.e. ⁢x∈Ω, where ⁢γ=q1⁢(2-q1ps*)-q12p.

3. There exists ϖ∈(p,ps*) such that

   0<ϖ⁢F⁢(x,t)⩽f⁢(x,t)⁢t for a.e. ⁢x∈Ω,with ⁢|t|⁢ small.

We will prove our results via a variational approach following the methods of [22, 6]. The strategy is to modify and extend f to an appropriate f~, and to show for the associated modified functional the existence of solutions with bounded L∞ norm, therefore to obtain solutions for the original problem (Pλ). So the L∞-estimate of the solution is very important. However, there are no Lp-estimates for fractional Laplace problems as the classic Laplace problem. Recently, in [12], Iannizzotto et al. proved a priori L∞ bounds on the weak solutions of problem (P1). By this estimate and the Sobolev embedding, we are able to get a more suitable estimate of L∞ norm of solutions and avoid further restriction on the behavior of f at infinity.

Throughout the paper, we denote by C various positive constants, whose values are not essential to the problem, and may be different from line to line. We denote the usual norm of Lq⁢(Ω) by ∥⋅∥q for 1⩽q⩽∞. Moreover, let 0<s<1<p<∞ be real numbers, and the fractional critical exponent be defined as

The paper is organized as follows. In Section 2, we introduce some preliminary notions and notations, and set the functional framework of our problem. In Section 3, we will prove Theorem 1.1. Section 4 is devoted to the proof Theorem 1.2. The proof of Theorem 1.5 will be given in Section 5.

2 Preliminary

In this preliminary section, for the reader’s convenience, we collect some basic results that will be used in the forthcoming sections.

Firstly, we introduce a variational setting for problem (Pλ). The Gagliardo seminorm is defined, for all measurable function u:ℝN↦ℝ, by

The fractional Sobolev space

is endowed with the norm

In this paper, we will work in the closed linear subspace

which can be equivalently renormed by setting ∥⋅∥=[⋅]s,p (see [8, Theorem 7.1]). It is readily seen that (X(Ω),∥⋅∥) is a uniformly convex Banach space and the following Sobolev embedding theorem holds.

In [12], the fractional p-Laplacian is redefined variationally as the nonlinear operator A:X⁢(Ω)↦X⁢(Ω)* defined, for all u,v∈X⁢(Ω), by

A weak solution of problem (Pλ) is a function of u∈X⁢(Ω) such that

Since X⁢(Ω) is uniformly convex, A satisfies the following compactness condition:

1. If {un} is a sequence in X⁢(Ω) such that un⇀u⁢in⁢X⁢(Ω) and 〈A⁢(un),un-u〉→0, then un→u⁢in⁢X⁢(Ω).

When f is odd in t, we need the following critical point theorem, see [22, Lemma 2.4].

3 Proof of Theorem 1.1

In this section we will prove Theorem 1.1. Since (f1)–(f3) give the behavior of f just in Ω×(-δ,δ), the functional ∫ΩF⁢(x,u)⁢𝑑x is not well-defined in X⁢(Ω). To overcome this difficulty, we need to modify and extend f to an appropriate f~, in the spirit of the arguments developed in [22]. For this purpose, we first observe that (f1) and (f2) imply, for small |t|, that

Let ρ⁢(t)∈C1⁢(ℝ,[0,1]) be an even cut-off function verifying t⁢ρ′⁢(t)⩽0 and

where 0<τ<δ2 is chosen such that (3.1), (3.2) and (f3) hold for |s|⩽2⁢τ. Set

It is easy to check that the following properties on f~ hold.

By Lemma 3.1, we can modify and extend f to get f~∈C⁢(Ω×ℝ,ℝ) satisfying all properties listed in Lemma 3.1. Now we define

Then I~λ⁢(u)∈C1⁢(X⁢(Ω),ℝ). We remark that f~⁢(x,t)=f⁢(x,t) for (x,t)∈Ω×[-τ,τ], and that a critical point u of I~λ is a solution of (Pλ) if and only if ∥u∥∞⩽τ.

Now we investigate the properties of the functional I~λ.

Lemma 3.2 implies that the trivial solution 0 of (Pλ) is the unique critical point of I~λ at the level 0.

Next we check that I~λ⁢(u) is coercive, i.e., I~λ⁢(u)→∞ as ∥u∥→∞, and I~λ satisfies the (PS) condition.

Now we are in the position to give the proof of Theorem 1.1.

4 Proof of Theorem 1.2

In this section, we deal with the case where f is odd near the origin. First of all, for any fixed λ>0, we will get the existence of the infinitely many critical points of the functional I~λ, by Lemma 2.3. By Lemmas 3.1 and 3.3, and (f2), we have in hand the facts that I~λ satisfies the (PS) condition, and that is even and bounded from below, with I~λ⁢(0)=0. Hence, it suffices to find, for any k∈ℕ, a subspace Xk and ρk>0 such that

For any k∈ℕ, we find k independent functions φi∈X⁢(Ω), i=1,…,k, and set Xk:=span⁡{φ1,φ2,…,φk}. By (f1) and the definition of F~⁢(x,t), we have F~⁢(x,t)⩾C⁢|t|p1 for t∈ℝ. Then

Since all norms on Xk are equivalent, p1<p, by choosing ρk>0 small enough, (4.1) holds. With all conditions of Lemma 2.3 being verified, we get a sequence of critical points unλ∈X⁢(Ω) with I~λ⁢(unλ)=cnλ→0 and cnλ<0 as n→∞. In particular, {unλ} is a (PS)0 sequence of I~λ and has a convergent subsequence, still denoted by {unλ}. By Lemma 3.2,