Multiplicity of solutions for nonlocal fractional equations with nonsmooth potentials ♦

. This paper is concerned a speciﬁc category of nonlocal fractional Laplacian problems that involve nonsmooth potentials. By utilizing an abstract critical point theorem for nonsmooth functionals and combining it with the analytical framework on fractional Sobolev spaces developed by Servadei and Valdinoci, we are able to establish the existence of at least three weak solutions for nonlocal fractional problems. Moreover, this work also generalizes and improves upon certain results presented in the existing literature.

A typical example for the Kernel K is given by K(x) = |x| −(n+2s) .In this case L K is the fractional Laplacian operator defined by These operators have various applications in different fields, including phase transitions, thin obstacles, finance, optimization, stratified materials, crystal dislocation, anomalous diffusion, semipermeable membranes, soft thin films, ultra-relativistic limits of quantum mechanics, multiple scattering, quasi-geostrophic flows, minimal surfaces, water waves, and materials science.For a basic introduction to this topic, we recommend referring to the references [1] and the monograph [2].
It is well-known that many free boundary problems and obstacle problems can be reduced to partial differential equations with nonsmooth potentials.The field of nonsmooth analysis is closely related to the development of critical points theory for nondifferentiable functions, particularly for locally Lipschitz continuous functionals based on Clarke's generalized gradient [18].This theory provides a suitable mathematical framework to extend the classic critical point theory for C 1 -functionals in a natural way, and to meet specific needs in applications such as nonsmooth mechanics and engineering.For a comprehensive understanding of this topic, we recommend referring to the monographs by [3,4,24] and references such as [8-13, 31-33, 38, 40], among others.
If F , G and H are differentiable, then problem (1.1) becomes into the following form In recent years, there have been many interesting results focusing on problem (1.2) using various methods.However, in our case, we only assume that the energy functional corresponding to problem 1.1 is locally Lipschitz instead of differentiable.This assumption poses certain difficulties and prevents us from applying classical variational methods to solve the problem.To overcome these difficulties, we need to utilize theories for locally Lipschitz functionals to establish existence results for this case.Fortunately, in [35,Theorem 3.3] (see Theorem 2.1 below), we have developed a nonsmooth three critical points theory that can be applied to prove that problem 1.1 has at least three critical points (see Theorem 3.1).One remarkable aspect of our results is that we do not impose any conditions on the behavior of the nonlinearities at the origin, which makes our results more interesting compared to most known results in the literature (e.g., [5-7, 36, 37] et.al.).
Recently, there has been significant attention focused on the study of fractional and nonlocal operators of elliptic type, both for pure mathematical research and with a view to concrete realworld applications.In [26], Servadel and Valdinoci proved the following fractional Laplacian equation: They proved a maximum principle and used it to obtain their regularity results.Autuori and Pucci [21] discussed the elliptic problems involving the fractional Laplacian in R N and derived three nontrivial critical values.Cabré and Sire [34] studied problem (1.3) and established necessary conditions on the nonlinearity f to admit certain types of solutions.In [22], Bisci, using variational methods, established three weak solutions via an abstract result by Ricceri about non-local equations.However, all of these works are based on the assumption that the potential functionals are smooth.To the best of our knowledge, there exist no results discussing problem (1.1) with nonsmooth potentials.For problems with nonsmooth potential functionals, most results focus on studying the Dirichlet problem involving the p-Laplacian or p(x)-Laplacian differential inclusion.For example, there exist some results studying the following problem with a nonsmooth potential in Sobolev spaces: − div(|∇u| p−2 ∇u) ∈ ∂F (x, u) for a.e.x ∈ Ω, Gansiński and Papageorgiou [24], using a variational approach combined with suitable truncation techniques and the method of upper-lower solutions, proved the existence of at least five nontrivial smooth solutions for problem (1.4).Iannizzotto and Marano [15], employing variational methods with truncation techniques, obtained at least three smooth solutions for problem (1.4) with ∂F (x, u) given by λ∂F (x, u).Besides, Kyritsi and Papageorgiou [30], based on the nonsmooth critical point theory of Chang [23], derived two strictly positive solutions with p ≥ 2. In [14], Kristály, employing a nonsmooth Ricceri-type variational principle, proved the existence of infinitely many, radially symmetric solutions of p-Laplacian differential inclusions in an unbounded domain.Results of p(x)-Laplacian differential inclusion can be found in [16,17,19,20,41,42].However, we should mention that the variational method to deal with problem (1.1) is not often easy to perform.Variational approaches do not work when applied to these classes of equations due to the presence of the nonlocal term.Fortunately, our approach in this paper is realizable by checking that the associated energy functional satisfies the assumptions requested by a very recent and general nonsmooth three critical points theorem derived by Yuan and Huang [35,Theorem 3.3] (see Theorem 2.1 below) and thanks to a suitable framework developed in [27].Furthermore, we observe a remarkable feature of our results: compared to most of the known results in the classical Laplacian case, no condition on the behavior of the involved nonlinearities at the origin is assumed.Therefore, our results are more interesting.
The rest of the paper is organized as follows.Section 2 contains the necessary preliminaries.In Section 3, we prove our main results.

Some basic notations
• means weak convergence, → strong convergence.
• C denotes all the embedding constants (the exact value may be different from line to line).
In this section, we briefly recall the definition of the functional space X 0 , firstly introduced in [28].The functional space X denotes the linear of Lebesgue measurable functions from R n to R such that the restriction to Ω of any function g in X belongs to L 2 (Ω) and where CΩ = R n \ Ω.We denote by X 0 the following linear subspace of X X 0 = {g ∈ X : g = 0 a.a. in R n \ Ω}.
Note that X and X 0 are non-empty, since C 2 0 (Ω) ⊆ X 0 by Lemma 11 in [28].Moreover, the space X is endowed with the norm defined as It is easy to see that .X is a norm on X (see, for instance [27]).By [27,Lemmas 6 and 7] we can take the function as norm on X 0 in the sequel.Also (X 0 , .X 0 ) is a Hilbert space with scalar product Lemma 7].
Note that in (2.1) (and in the related scalar product) the integral can be extended to all R n × R n , since v ∈ X 0 (and so v = 0 a.a. in R n \ Ω).While for a general kernel K satisfying conditions from (K 1 ) to (K 3 ) we have that the space X 0 consists of all the functions of the usual fractional Sobolev space H s (R n ) which vanish a.a.outside Ω (see [29,Lemma 7]).
Here H s (R n ) denotes the fractional Sobolev space endowed with the norm (the so-called Gagliardo norm) Recall the embedding properties of X 0 into the usual Lebesgue spaces (see [27,Lemma 8]).The embedding j : X 0 → L q (R n ) is continuous for any q ∈ [1, 2 * ], while it is compact when q ∈ [1, 2 * ), where 2 * = 2n n−2s denotes the fractional critical Sobolev exponent.Hence, for any q ∈ [1, 2 * ] there exists a positive constant c q such that In what follows, let λ 1 be the 1-th eigenvalue of the operator −L K with homogenous Dirichlet boundary data, namely the 1-th eigenvalue of the problem Note that, as in the classical Laplacian case, the set of the eigenvalues of problem (2.2) consists of a sequence {λ k } k∈N with It is easy to see that the function ν → I 0 (u; ν) is sublinear, continuous and so is the support function of a nonempty, convex and w * − compact set ∂I(u) ⊂ X * , defined by Clearly, these definitions extend those of the Gâteaux directional derivative and gradient.
A point u ∈ X is a critical point of I, if 0 ∈ ∂I(u).It is easy to see that, if u ∈ X is a local minimum of I, then 0 ∈ ∂I(u).For more details we refer the reader to Clarke [18].
(iii) Let j : X → R be a continuously differentiable function.Then ∂j(u) = {j (u)}, j • (u; z) coincides with j (u), z X and (h + j) • (u; z) = h • (u; z) + j (u), z X for all u, z ∈ X; (iv) (Lebourg's mean value theorem) Let u and v be two points in X.Then, there exists a point ω in the open segment between u and v, and (v) Let Y be a Banach space and j : Y → X a continuously differentiable function.Then h • j is locally Lipschitz and (vii) ∂h(u) is convex and weakly * compact and the set-valued mapping ∂h : X → 2 X * is weakly * upper semicontinuous; (viii) ∂(λh)(u) = λ∂h(u) for every λ ∈ R.
Let I, Ψ, Φ : X → R be three given functions, for each µ > 0 and r ∈] inf X Φ, sup X Φ[, we set and

I(u).
With the above notations, our abstract tool for proving the main result of our paper is [35, Theorem 3.3] and we recall here for the readers' convenience.
Theorem 2.1 Let (X, • ) be a reflexive Banach space, I ∈ C 1 (X, R) a sequentially weakly lower semicontinuous function, bounded on any bounded subset of X, such that I is of type (S) + .Ψ and Φ : X → R are two locally Lipschitz functions with compact gradient.Assume also that the function Ψ + λΦ is bounded below for all λ > 0 and that lim inf Then, for each r > sup N Φ, where N is the set of all global minima of I, for each µ > max{0, h 3 (I, Ψ, Φ, r)} and each compact interval [a, b] ⊂]0, h 2 (µI + Ψ, Φ, r)[, there exists a number ρ > 0 with the following property: for every λ ∈ [a, b] and every locally Lipschitz function H : X → R with compact gradient, there exists δ > 0 such that, for each ν ∈ [0, δ], the function µI(u) + Ψ(u) + λΦ(u) + νH(u) has at least three critical points in X whose norms are less than ρ.

The main results
Firstly, we define I(u), Ψ(u), Φ(u), H(u) : X 0 → R by for all u ∈ X.It is easy to see that the functional I is a continuously Gâteaux differentiable whose Gâteaux derivative at the point u ∈ X 0 is the functional I (u) ∈ X * 0 given by and ηr = inf For each ∈ 0, In order to discuss problem (1.1), we need the following hypotheses: where α ∈ (2, 2 * ); where α ∈ (2, 2 * ); Remark 3.1 It is easy to see that there exist lots of functions which satisfy hypotheses (F 1 )-(F 4 ), (G 1 )-(G 4 ) and (H 1 )-(H 4 ).For example, for simplicity, we drop the x-dependence. and I : X 0 → X * 0 is a continuous, bounded and strictly monotone operator; (ii) I is a mapping of type (S + ), i.e., if u n u in X 0 and Proof.(i) By virtue of the properties of (K 1 ) − (K 3 ), it is obvious that I is continuous and bounded.Note that then, we have i.e., I is monotone.In fact I is strictly monotone.Indeed, if This means that I is a strictly monotone operator in X.
(ii) From (i), if u n u and lim n→+∞ I (u) − I (v), u − v ≤ 0, then lim n→+∞ I (u) − I (v), u − v = 0.According to (3.1), u n → u in Ω, so we obtain a subsequence (which we still denoted by u n ) satisfying u n → u a.a.x ∈ Ω.From Fadou's lemma, we have By u n u we derive On the other hand, we also have Therefore, u n → u in X 0 , i.e., I is of type (S + ).
The next Lemma displays some properties of F (u).
Lemma 3.2 If hypotheses (F 1 ) − (F 3 ) hold, then F : X → R is a locally Lipschitz function with compact gradient.
Proof.Firstly we show that F is locally Lipschitz.Let u, v ∈ X 0 .According to the Lebourg's mean value theorem, we have Then it is easy to see that F is locally Lipschitz.
Next, we prove that ∂F is compact.Choosing u ∈ X 0 , u * ∈ ∂F (u) we obtain that for every and F • (u; •) : L r (Ω) → R is a subadditive function (see Proposition 2.1).Furthermore, u * ∈ X * 0 is continuous also with respect to the topology induced on X 0 by the norm • L r (Ω) .Indeed, setting L > 0 a Lipschitz constant for F in a neighborhood of u, for all z ∈ X 0 we derive from Proposition 2.1 (ii) Hence, by Hahn-Banach Theorem, u * can be extended to an element of the dual L r (Ω) (complying with (3.5)) for all v ∈ L r (Ω), this means that we can represent u * as an element of L r (Ω) and write for every v ∈ L r (Ω) Set {u n } be a sequence in X 0 such that u n ≤ M for all n ∈ N (M > 0) and take ξ n ∈ ∂F (u n ) for all n ∈ N. It follows from (F 3 ) and (3.6) that 1 ), i.e., the sequence {ξ n } is bounded.Hence, passing to a subsequence, we have ξ n ξ ∈ X * 0 .We will prove that {ξ n } ⊂ X * 0 has a strong convergence.We proceed by contradiction.Assume that there exists some ε > 0 such that ξ n − ξ X * 0 > ε for all n ∈ N and hence for all n ∈ N there exists v n ∈ B(0, 1) such that Recall that {v n } is a bounded sequence and passing to a subsequence, one has Hence, for n large enough, we have Then, ≤ ε, which contradicts to (3.7).
Similar, we have the following properties of the functions Φ(u) and H(u).With the above lemmas, our main result reads as follows.