Differential inclusion problems with convolution and discontinuous nonlinearities

The paper investigates a new type of differential inclusion problem driven by a weighted (p, q)-Laplacian and subject to Dirichlet boundary condition. The problem fully depends on the solution and its gradient. The main novelty is that the problem exhibits simultaneously a nonlocal term involving convolution with the solution and a multivalued term describing discontinuous nonlinearities for the solution. Results stating existence, uniqueness and dependence on parameters are established.


1.
Introduction. The aim of this paper is to study the quasilinear differential inclusion problem − ∆ p u − µ(x)∆ q u ∈ f (x, ρ * u, ∇(ρ * u)) + [g(u), g(u)] in Ω on a bounded domain Ω ⊂ R N , for N ≥ 2, with the boundary ∂Ω. In order to simplify the presentation, we assume from the beginning that p < N . The case p ≥ N is actually simpler and can be handled along the same lines. As can be seen from the statement, (1) is a nonstandard problem, in particular it is nonlocal, for which the meaning of the data are described below. Here ∆ p : W 1,p 0 (Ω) → W −1,p (Ω) and ∆ q : W 1,q 0 (Ω) → W −1,q (Ω), with 1 < q < p < +∞, p = p p−1 and q = q q−1 , are the p-Laplacian and the q-Laplacian, respectively. The leading operator in problem (1) is the (negative) weighted (p, q)-Laplacian −∆ p − µ(x)∆ q with the nonnegative weight µ ∈ L ∞ (Ω). An important case is for µ = 0 with the (negative) p-Laplacian −∆ p as driving operator. Another important case is for µ = 1, where the driving operator is the (negative) (p, q)- The right-hand side of (1) is described by a Carathéodory function f : Ω × R × R N → R (i.e., f (·, s, ξ) is measurable on Ω for all (s, ξ) ∈ R × R N and f (x, ·, ·) is continuous for a.e. x ∈ Ω). In (1), f (x, ρ * u, ∇(ρ * u)) is a convection term, i.e., it depends on the solution u and its gradient ∇u. In fact, it is much more than a convection term because it involves a function ρ ∈ L 1 (R N ) (that can be seen as a parameter) through the convolution ρ * u with the solution u ∈ W 1,p 0 (Ω). Due to the convolution, in order to have a meaningful formulation it is convenient to consider the Sobolev space W 1,p 0 (Ω) embedded in W 1,p (R N ) by identifying every u ∈ W 1,p 0 (Ω) with its extensionũ ∈ W 1,p (R N ) equal to zero outside Ω. In this way we have the inclusion map E : W 1,p 0 (Ω) → W 1,p (R N ) given by the extension with 0 outside Ω, so E(u) =ũ. Then for ρ ∈ L 1 (R N ) and u ∈ W 1,p 0 (Ω) ⊂ W 1,p (R N ) the convolution ρ * u is defined by Notice that it holds supp ρ * u ⊂ Ω + supp ρ. The gradient ∇(ρ * u) appearing in (1) makes sense taking into account that ρ * u = ρ * ũ ∈ W 1,p (R N ).
If the function g is continuous, then the interval [g(u(x)), g(u(x))] collapses to the singleton g(u(x)). Consequently, in this case (1) reduces to the quasilinear Dirichlet equation in Ω, involving convection and convolution. The multivalued term [g(u), g(u)] in (1) is actually the generalized gradient of a locally Lipschitz function as will be shown explicitly in the next section. This fact qualifies problem (1) as a hemivariational inequality, which is of a special type due to the convection term f (x, ρ * u, ∇(ρ * u)) exhibiting composition with convolution. Besides their substantial mathematical interest, the hemivariational inequalities represented a major progress by passing from convex nonsmooth potentials to nonconvex nonsmooth potentials in order to model complicated phenomena with various contact laws in mechanics and engineering. For many results and applications in this direction we refer to [6,9,10,12,13], It is worth mentioning that problems (1) and (4) do not have variational (smooth or nonsmooth) structure, so the variational methods are not applicable. This causes considerable technical difficulties in dealing with such problems. Generally, the variational structure is lost when there is a convection term since the presence of the gradient ∇u in the right-hand side prevents the construction of potential. Here the situation is even more difficult because of the composition with the convolution ρ * u, which is a nonlocal operator. If f = 0, problem (1) becomes a nonsmooth variational problem with discontinuous nonlinearities (see [4,9]), while (4) is a quasilinear elliptic equation that can be treated by using the smooth critical point theory. If g = 0, problems (1) and (4) reduce to those considered in [8] whose results are extended in the present work.
The main novelty of the paper is that we are able to deal simultaneously with various top difficulties: weighted (p, q)-Laplacian, convection, convolution, discontinuous nonlinearity and multivalued term. Under verifiable conditions we prove existence of solutions. Strengthening the hypotheses, we provide a uniqueness result, too. Moreover, we also investigate the dependence of the solution set to problems (1) and (4) with respect to µ ∈ L ∞ + (Ω) and ρ ∈ L 1 (R N ) considered as parameters. Results on dependence regarding ρ ∈ L 1 (R N ) for problems without multivalued terms and discontinuous nonlinearities are given in [8]. Results on dependence regarding µ ∈ R for problems without convolution, multivalued terms and discontinuous nonlinearities are given in [1] (see also [7]).
The rest of the paper is organized as follows. Section 2 is devoted to the mathematical background needed in the sequel. Section 3 contains our existence result. Section 4 presents our uniqueness result. Section 5 focuses on the dependence with respect to parameters.
2. Mathematical background. This section provides the necessary mathematical background for our results on problem (1), in particular (4).
We start by briefly reviewing the multivalued pseudomonotone operators. More details can be found in [3,11,14]. Let X be a reflexive Banach space with the norm · , its dual X * and the duality pairing ·, · between X and X * . The norm convergence in X and X * is denoted by →, while the weak convergence is denoted by . A multivalued map A : X → 2 X * is called bounded if it maps bounded sets into bounded sets. It is said to be coercive if there is a function c : R + → R with c(t) → +∞ as t → +∞ such that ξ, u − u 0 ≥ c( u ) u for all ξ ∈ A(u) and some u 0 ∈ X. A multivalued map A : X → 2 X * is called pseudomonotone if (i) for each v ∈ X, the set Av ⊂ X * is nonempty, bounded, closed and convex; (ii) A is upper semicontinuous from each finite dimensional subspace of X to X * endowed with the weak topology; (iii) for any sequences (u n ) ⊂ X and (u * n ) ⊂ X * satisfying u n u in X, u * n ∈ Au n for all n and lim sup n→∞ u * n , u n − u ≤ 0, and for each v ∈ X there exists u * (v) ∈ Au such that We recall the main theorem for pseudomonotone operators (see, e.g., [3, Theorem 2.125]).
Theorem 2.1. Let X be a reflexive Banach space, let A : X → 2 X * be a pseudomonotone, bounded and coercive operator, and let η ∈ X * . Then there exists at least one u ∈ X with η ∈ Au.
Next we outline some basic elements of nonsmooth analysis related to locally Lipschitz functions. An extensive study of this topic can be found in [5,4,9,13]). A function Φ : X → R on a Banach space X is called locally Lipschitz if for every u ∈ X there is a neighborhood U of u in X and a constant L u > 0 such that The generalized directional derivative of a locally Lipschitz function Φ : X → R at u ∈ X in the direction v ∈ X is defined as and the generalized gradient of Φ at u ∈ X is the subset of the dual space X * given by It is useful to point out that a continuous and convex function Φ : X → R is locally Lipschitz and its generalized gradient ∂Φ : X → 2 Y * coincides with the subdifferential of Φ in the sense of convex analysis. As another important example, if Φ : X → R is a continuously differentiable function, the generalized gradient of Φ is just the differential DΦ of Φ. The preceding notions of subdifferentiability theory for locally Lipschitz functions are needed to handle the multivalued term [g(u), g(u)] in problem (1) in the way as explained in the following. Given g : The function G : R → R is locally Lipschitz and one can show that the generalized gradient ∂G(s) of G at any s ∈ R is the compact interval in R determined by where g(s) and g(s) are precisely the functions in (2) and (3), respectively (see, e.g., [5, Example 2.2.5]). Now we describe the functional setting for problems (1) and (4). We suppose that 1 < q < p < +∞ and consider the Sobolev spaces W 1,p 0 (Ω) and W 1,q 0 (Ω) endowed with the norms u := ∇u p and ∇u q , respectively, where · r stands for the usual L r -norm. The duals of the spaces W 1,p 0 (Ω) and W 1,q 0 (Ω) are W −1,p (Ω) and W −1,q (Ω), respectively. As usual, we denote by p * the Sobolev critical exponent, that is p * = N p/(N − p) (recall that we assume p < N ). As p ∈ (1, N ), Rellich-Kondrachov theorem asserts that W 1,p 0 (Ω) is compactly embedded into L θ (Ω) if 1 ≤ θ < p * and continuously embedded for θ = p * . Thus for every r ∈ [1, p * ] there exists a positive constant S r such that The (negative) p-Laplacian −∆ p : Among many properties of this nonlinear operator, we mention that −∆ p is strictly monotone and continuous, so pseudomonotone. If p = 2 it is just the ordinary (negative) Laplacian operator −∆. Similarly, we have the definition of the (negative) q-Laplacian −∆ q : W 1,q 0 (Ω) → W −1,q (Ω). By virtue of the embedding W 1,p 0 (Ω) → W 1,q 0 (Ω), there exists a constant k > 0 such that Then the differential operator −∆ p − µ(x)∆ q driving inclusion (1) and equation (4) is well defined on W 1,p 0 (Ω) since µ ∈ L ∞ (Ω). Moreover, taking into account that the sum of pseudomonotone operators is pseudomonotone, the nonlinear operator −∆ p − µ(x)∆ q is pseudomonotone, actually maximal monotone because it was supposed that µ(x) ≥ 0 for a.e. x ∈ Ω.
3. Existence of solutions. In this section we focus on the existence of solutions to problems (1) and (4). The hypotheses that we assume are the following: (H g ) The function g : R → R is measurable and there exist constants c > 0 and σ ∈ (1, p) such that Condition (H f ) has been used in [8].
Our existence result on problems (1) and (4) is as follows.
Theorem 3.1. Assume that conditions (H f ) and (H g ) hold. Then problem (1) admits at least one solution. In particular, if the function g is continuous, then a solution to problem (4) exists.
Recall the function G : R → R in (5) corresponding to g : R → R in the righthand side of (1). Assumption (H g ) implies that G : R → R is locally Lipschitz. Then the functional Φ : L σ (Ω) → R given by is Lipschitz continuous on the bounded subsets of L σ (Ω). Hence we can see it as a locally Lipschitz function on W 1,p 0 (Ω) (note that σ < p) and its generalized gradient can be regarded as a multivalued map ∂Φ : . This multivalued operator is bounded thanks to the fact that Φ is Lipschitz continuous on the bounded subsets of L σ (Ω).
Due to the presence of convolution, it is convenient to identify f : Ω×R×R N → R with the functionf : R N ×R×R N → R obtained by extending f (·, s, ξ) by 0 outside Ω. Then the growth condition in assumption (H f ) ensures that the Nemytskii operator N f : is well defined, continuous and bounded. Using the inclusion map E : W 1,p 0 (Ω) → W 1,p (R N ) obtained by taking the zero extension outside Ω (see Section 1) and its adjoint map E * : The preceding comments clarify that the multivalued operator A : W 1,p 0 (Ω) → 2 W −1,p (Ω) is bounded.

4.
A uniqueness result. Our objective in this section is to provide sufficient conditions to have a unique solution to problems (1) and (4). Basically, this occurs under additional assumptions of Lipschitz-like conditions.
(b) The existence of a solution is guaranteed by Theorem 3.1. Let u 1 , u 2 ∈ W 1,p 0 (Ω) be solutions to problem (1) with q = 2 < p < +∞. Proceeding as in part (a), we can rely on (13), (14), (26) and (27) to see that Then arguing as in part (a) we are led to Now it suffices to address hypothesis (29) for obtaining u 1 = u 2 . The proof is thus complete.
In the case of problem (4) with g continuous, Theorem 4.1 takes the form.
Proof. The corollary follows readily from Theorem 4.1 noticing that, under the hypothesis of continuity for the function g, condition (27) reads as (32), which is apparent from (2) and (3). Remark 1. If the function g : R → R is nonincreasing or Lipschitz continuous, then condition (32) is fulfilled.

Dependence on parameters.
In this section we investigate the dependence of the solutions to problems (1) and (4) with respect to the parameters µ ∈ L ∞ + (Ω) and ρ ∈ L 1 (R N ).
The following statement deals with the dependence on the parameter ρ ∈ L 1 (R N ) in problem (1).
Let us use u n as test function in (33). Then (34) and (35) imply that the sequence (u n ) is bounded in W 1,p 0 (Ω), so up to a subsequence u n u in W 1,p 0 (Ω) for some u ∈ W 1,p 0 (Ω). Along a reasoning similar to the one developed in the proof of Theorem 3.1, this time essentially based on the boundedness of the sequence (ρ n ) in L 1 (R N ), we are able to reach (24). From now on we can continue as in the proof of Theorem 3.1 achieving the strong convergence u n → u in W 1,p 0 (Ω). Here the validity of the (S + )-property of the operator −∆ p − µ(x)∆ q on W 1,p 0 (Ω) as checked in the mentioned proof is crucial. We now note that (10) and (11) yield the strong convergence ρ n * u n → ρ * u in W 1,p (R N ). Moreover, passing to a relabeled subsequence we may suppose that u n (x) → u(x), ρ n * u n (x) → ρ * u(x) and ρ n * ∇u n (x) → ρ * ∇u(x) for a.e. x ∈ Ω.
Recall that u n solves (33) which by means of (6) reads as On the basis of the continuity of the operators −∆ p −µ(x)∆ q and N f in conjunction with the strong convergence u n → u in W 1,p 0 (Ω) and the strong convergence ρ n * u n → ρ * u in W 1,p (R N ), besides the graph closedness of the generalized gradient ∂G of the locally Lipschitz function G in (5), we can pass to the limit in the above inequality as n → +∞ obtaining through Fatou's lemma that u is a solution of (1). The proof is thus complete.
This together with the weak convergence u n u in W 1,q 0 (Ω) permits to invoke the (S + )-property of the operator −∆ q : W 1,q 0 (Ω) → W −1,q (Ω) for obtaining the strong convergence u n → u in W 1,q 0 (Ω). Based on the previous analysis, we can pass to the limit in (40) leading to ∆ q u = 0, which yields u = 0. The proof is thus complete.