Existence and multiplicity of solutions for fractional p-Laplacian equation involving critical concave-convex nonlinearities

1 Introduction

In this paper, we are interested in the following fractional p-Laplacian equation

where s ∈ (0, 1), p > q > 1, n > sp, λ > 0, p s * = n p n − s p is called the critical Sobolev exponent, and Ω ⊂ R n is a bounded domain with C ^1,1 boundary. To give the weak formulation of ( − Δ ) p s , we denote the Gagliardo seminorm by

Let

be endowed with the norm

where |⋅| _p stands for the usual norm of L p ( R n ) . Denote the subspace

equivalently renormed with u = [ u ] s , p (see [1, Theorem 7.1]), it is well known that W 0 s , p ( Ω ) is a uniformly convex Banach space. Furthermore, the embedding W 0 s , p ( Ω ) ↪ L r ( Ω ) is continuous for r ∈ 1 , p s * and compact for r ∈ [ 1 , p s * ) , see [1, Theorems 6.5, 7.1]. ( − Δ ) p s can be variationally regarded as an operator from W 0 s , p ( Ω ) into its dual space W 0 s , p ( Ω ) * as follows,

where J _u (x, y) = |u(x) − u(y)| ^p−2(u(x) − u(y)). Let f : Ω × R → R be a Carathéodory mapping, and consider the general fractional p-Laplacian equation

We call u ∈ W 0 s , p ( Ω ) a weak solution (respectively weak subsolution, weak supersolution) of (1.1) if f ( x , u ) ∈ W 0 s , p ( Ω ) * and

For simplicity, from now on, we omit the term weak in the rest of our paper. If f satisfies the following growth condition: for some C 0 > 0,1 < r ≤ p s * ,

then the functional

is C ¹ over W 0 s , p ( Ω ) , and any critical point of E is a solution to (1.1).

In order to find critical points of E, the eigenvalues of ( − Δ ) p s based on Z 2 -cohomological index introduced in [2] are usually used to carry out some linking constructions, see for instance [3], [4], [5] and references therein.

When f satisfies the subcritical growth condition (i.e. (1.2) with 1 < r < p s * ), many results about existence and multiplicity of solutions of (1.1) have been established, see for example [4], [6], [7], [8] and references therein.

In the special case p = 2, Ros-Oton and Serra [9] showed a Pohozaev identity for solutions of (1.1). When p ≠ 2, a similar identity for W 0 s , p ( Ω ) solution to (1.1) was conjectured in [4, Section 7], which could lead to the non-existence of positive solution to (1.1) on star-shaped Ω with f ( x , t ) = | t | p s * − 2 t . However, as far as we are aware, the validity of Pohozaev identity is still open for ( − Δ ) p s with p ≠ 2. This motivates people to consider the perturbation problem

where g : Ω × R → R has the subcritical growth, i.e. (1.2) with 1 < r < p s * .

It is well known that the associated energy functional of (1.4) cannot satisfy Palais–Smale condition globally, because of the non-compactness of the embedding W 0 s , p ( Ω ) ⊂ L p s * ( Ω ) . However, as Brezis and Nirenberg remarked in [10], the Palais–Smale condition can hold true under suitable thresholds related to the best Sobolev constant

where D s , p ( R n ) ≔ u ∈ L p s * ( R n ) : [ u ] s , p < ∞ .

Since Brezis-Nirenberg’s milestone work [10], a huge literature exists for the study of problem (1.4), here we will concentrate on the convex-concave situation combined with the critical Sobolev exponent. In the classical case with s = 1, (P _λ ) reads

where n > p > q > 1, p * = n p n − p and Δ _p u = div(|∇u| ^p−2∇u).

Ambrosetti, Brezis and Cerami [11] did a seminal work with the Laplacian, i.e. for p = 2 in (1.6). They proved that there exists Λ > 0 such that

1. Problem (1.6) has at least one positive solution if 0 < λ < Λ;

2. Problem (1.6) has at least one positive solution if λ = Λ;

3. Problem (1.6) has no positive solution if λ > Λ.

   Moreover, using mountain pass theorem, they obtained that

4. Problem (1.6) has at least two positive solutions if 0 < λ < Λ.

García Azorero, Manfredi and Peral Alonso [12] (see also [13]) generalized the above results (i)–(iii) to the quasilinear case with p-Laplacian, i.e. s = 1 and p > 1. They showed the result (iv) with

For the linear fractional Laplacian case, that is p = 2, s ∈ (0, 1), n > 2s and 1 < q < 2, the above results (i)–(iv) for (P _λ ) were obtained by Barrios, Colorado, Servadei and Soria [14, Theorem 1.1], see also [15]. For more discussions on fractional convex-concave or critical exponent problems, we refer to [16], [17] and references therein.

For general fractional p-Laplacian, the problem (P _λ ) is much less understood and few results exist. In [18], Mahwin and Molica Bisci showed that there exists an interval V ⊂ ( 0 , ∞ ) such that for every λ ∈ V , (1.4) admits at least one solution with general subcritical g, which is a local minimizer of the corresponding energy. The main strategy in [18] is to check the weak lower semicontinuity of the functional

restricted in a sufficiently small ball of W 0 s , p ( Ω ) . In particular, they showed that, see [18, Theorem 1.2],

has a positive solution for λ ∈ V , provided 2 ≤ q < p < r < p s * .

Motivated by the works mentioned above, we are concerned here with the existence and multiplicity issue for (P _λ ) with general p > q > 1, s ∈ (0, 1). Let d_Ω(x) ≔ dist(x, ∂Ω), we will use the following weighted space

endowed with the norm

Our first result is a dichotomy claim.

The novelty is multifold. Firstly, our result extends [12], [14] to general fractional p-Laplacian convex-concave problems (P _λ ) for all p > q > 1 and 0 < s < 1. Secondly, in [12], it was shown that (1.6) has an extremal solution u _Λ in the distributional sense, but no Sobolev regularity of u _Λ was mentioned. With similar proof to Theorem 1.1, we can claim that u _Λ in [12, Lemma 6.3] belongs to W 0 1 , p ( Ω ) . Finally, our approach works for more general nonlinearities g(x, u), for example to (1.7) with 1 < q < p < r < p s * , hence generalizes [18, Theorem 1.2], we leave the details for interested readers.

The fractional p-Laplacian brings notable differences and many difficulties comparing to previous works. In classical case with Laplacian, Ambrosetti, Brezis and Cerami [11] used super-subsolution method to find the minimal positive solution u _λ for λ ∈ (0, Λ), which is stable (i.e. the second variation of the energy functional at u _λ is nonnegative). The stability yields then {u _λ } is bounded in H 0 1 ( Ω ) , which leads to a weak solution to (1.6) for the limiting case λ = Λ.

In contrast to [11], we cannot apply Picone’s identity here due to the presence of the nonlocal operator, and there is no related stability theory for fractional p-Laplacian so far. Our idea is to consider the associated problem

We will show that (Q _λ ) has a unique solution v _λ which is less than any positive solution to (P _λ ). The key observation is that any positive solution to (1.1) can be controlled by d Ω s ( x ) , see Proposition 2.9 below. This permits us to construct suitable sub or supersolutions, and apply the weak comparison principle (see Lemma 2.5 below) to handle the uniqueness issue for (Q _λ ). We apply then the iteration method to compare v _λ with positive solution to (P _λ ), and deduce the existence of minimal positive solution u _λ to (P _λ ).

To show the existence of extremal solution to (P _Λ), the main difficulty is to show the boundedness of u _λ in W 0 s , p ( Ω ) . For that, we consider

Our aim is to find a local minimum solution u ̂ λ ∈ Σ with respect to the topology of C s 0 ( Ω ̄ ) , and use then the stability in C s 0 ( Ω ̄ ) to get the uniform bound of u ̂ λ in W 0 s , p ( Ω ) . For that, we need to show Σ is non empty, and u _λ′, u _λ″ are separated in C s 0 ( Ω ̄ ) topology. The essential difficulty here is the lack of strong comparison principle for ( − Δ ) p s as follows:

In [19], the author derived a version of strong comparison principle with rather restrictive assumptions, hence not applicable here. We will use the scale method (see Lemma 3.3 below) to obtain the strong comparison between u _λ′ and u _λ″. Finally, the boundedness of u ̂ λ in W 0 s , p ( Ω ) yields the boundedness of minimal solutions u _λ in W 0 s , p ( Ω ) .

Furthermore, we consider the existence of a second positive solution to (P _λ ), which is new in the frame of fractional p-Laplacian.

We use the variational method as in previous works, but owing to the nonlinear and nonlocal properties of ( − Δ ) p s , the setting and understanding of the mountain pass structure are much more involved than [11], [12], [14].

By maximum principle (see Lemma 2.7 below), positive solutions to (P _λ ) coincide with nontrivial critical points of the following functional in W 0 s , p ( Ω )

where u ⁺ = max{u, 0}.

One major argument in [11] for finding two positive solutions is due to Brezis and Nirenberg [20], which connects variational and nonvariational methods. Roughly speaking, they showed that a local minimizer in C ¹ topology is also a local minimizer in H 0 1 ( Ω ) . In fractional Laplacian case, C s 0 ( Ω ̄ ) can serve as a suitable substitute of C 1 ( Ω ̄ ) . Iannizzotto, Mosconi and Squassina [21] proved that for p ≥ 2 and f satisfying (1.2), a local minimizer of E (see (1.3)) in C s 0 ( Ω ̄ ) ∩ W 0 s , p ( Ω ) with respect to C s 0 ( Ω ̄ ) topology is also a local minimizer in W 0 s , p ( Ω ) , see Theorem 2.10 below.

To prove Theorem 1.2, we show first I ̃ λ has a local minimizer u ̂ λ in Σ (see (1.9)) with respect to C s 0 ( Ω ̄ ) topology, and prove then it is also a local minimizer in W 0 s , p ( Ω ) when p ≥ 2. We can assume u ̂ λ = u λ , the minimal solution of (P _λ ) provided by Theorem 1.1, otherwise two positive solutions occur already. Under the assumption that I ̃ λ has only two critical points 0 and u _λ , we will show (see Proposition 4.3) that I ̃ λ satisfies the Palais–Smale condition for all levels

where S _s,p is the best Sobolev constant in (1.5). It remains now to construct a mountain pass geometry of I ̃ λ around u _λ , and check that the mountain pass level satisfies (1.11).

In the classical case as [11], we use currently the minimizers for the Sobolev embedding to handle the mountain pass level. For general s ∈ (0, 1), p > 1, it has been conjectured in [22] that the minimizers for S _s,p are of the form cU(|x − x ₀|/ɛ), where

As far as we are aware, this conjecture remains open. Only the constant sign, asymptotic decay and radial symmetry for minimizers were established in [22, Theorem 1.1]. To estimate the mountain pass level, we will make use of some truncation functions u _ɛ,δ constructed by Mosconi et al. [5], see Section 2.1 below.

The mountain pass setting with ( − Δ ) p s is also more involved. In contrast to [5], our mountain pass geometry is around the minimal solution u _λ instead of 0, for which the estimates will be more complex, we need to proceed carefully and simultaneously the construction of mountain pass geometry and the estimate for mountain pass level.

To be more precise, we consider η _δ u _λ where η _δ is a cut-off function, see (2.6). We choose u _λ as the starting point of mountain pass path, with the terminal point

where t ₀ > 0 (depending on ɛ and δ) is chosen such that I ̃ λ ( e ) < I ̃ λ ( u λ ) . Consider the set of mountain pass paths

and the corresponding mountain pass level

We will select a special path

and check that I ̃ λ ( γ ε , δ ( t ) ) tends to I ̃ λ ( u λ ) < c s , p as δ → 0 for all 0 ≤ t ≤ 1 2 . To reach m _ɛ,δ < c _s,p , we will prove

for some ɛ, δ > 0. To handle many involved integral estimates, a key argument here is that u _ɛ,δ and η _δ u _λ have disjoint support domains. Under the assumptions of Theorem 1.2, taking ɛ = δ ^k+1 with suitable choice of k ∈ (0, p − 1), we can claim that (1.16) holds whenever δ is sufficiently small. The mountain pass technique provides then a second critical point. Finally, we prove that the mountain pass level is positive for λ > 0 but small, which guarantees the non-triviality of second critical point.

Note also that our estimates in Section 4 are novel, and different from the previous works [11], [12], [14], we need two parameters to control the order of the blow-up.

Finally, without sign constraint, we obtain infinitely many solutions for (P _λ ). Here we consider

Bhakta and Mukherjee [23] used Nehari manifold to show the existence of at least one sign changing solution to (P _λ ), see also [24], [25] for systems with fractional p-Lapalcian and critical nonlinearities.

For the p-Laplacian case, i.e. with s = 1, García Azorero and Peral Alonso in [26, Theorem 4.5] proved the existence of infinitely many solutions of (1.6), provided λ > 0 is small, 1 < q < p and n > p (see also [11] for p = 2). They used Z 2 -genus and Lusternik–Schnirelman theory (see for instance [27, Theorem 10.9]). Similar result was shown also with p = 2, based on the dual Fountain theorem due to Bartsch and Willem [28] (see also [29]).

It is worthy to mention that the classical dual Fountain theorem may not work directly for general Banach spaces, since there is no natural decomposition such that the projection operator approaches the identity. We provide a space decomposition for reflexible and separable Banach space, see Theorem 5.2 below.

Here we use the Z 2 -genus to construct a sequence of minimax levels b _j for the functional I _λ in a small ball B _r (0) of W 0 s , p ( Ω ) , see (5.1) below. We will show that b _j are negative critical values of I _λ , and lower bounded by another family of special mountain pass levels b ̃ j (see (5.4)). By means of weak convergence argument (see Theorem 5.2 (ii)), we prove that for small enough r > 0, b ̃ j tends to 0⁻, so does b _j .

The paper is organized as follows. In Section 2, we introduce some notations and preliminary results. The proof of Theorems 1.1, 1.2 and 1.4 are completed respectively in Sections 3–5.

2 Notations and preliminaries

In this paper, we use the following notations.

1. C, C′, C ₁, C ₂, … denote always generic positive constants.

2. |⋅| _p means the usual norm of L p ( R n ) or L ^p (Ω).

3. ⋅ denotes the norm of W 0 s , p ( Ω ) .

4. For any v ∈ W 0 s , p ( Ω ) , J _v (x, y) = |v(x) − v(y)| ^p−2(v(x) − v(y)).

5. We use φ ₁ to denote the eigenfunction corresponding to the first eigenvalue λ ₁ of ( − Δ ) p s in W 0 s , p ( Ω ) such that φ ₁ > 0 in Ω and |φ ₁|_∞ = 1.

6. S _s,p denotes the best Sobolev constant defined in (1.5).

7. The function space C s 0 ( Ω ̄ ) is given in (1.8).

3 Proof of Theorem 1.1

For simplicity, we will not repeat always the regularity assumption for Ω, also we often omit to repeat n > sp. Even the procedures look similar to [11], we need more involved argument for each step. Consider

Inspired by [12], we first check that Λ is finite and positive.