SOLUTIONS FOR NONLINEAR VARIATIONAL INEQUALITIES WITH A NONSMOOTH POTENTIAL

First we examine a resonant variational inequality driven by the p-Laplacian and with a nonsmooth potential. We prove the existence of a nontrivial solution. Then we use this existence theorem to obtain nontrivial positive solutions for a class of resonant elliptic equations involving the p-Laplacian and a nonsmooth potential. Our approach is variational based on the nonsmooth critical point theory for functionals of the form φ= φ1 +φ2 with φ1 locally Lipschitz and φ2 proper, convex, lower semicontinuous.


Introduction
In this paper, we consider the following nonlinear variational inequality at resonance with a nonsmooth potential function (Z ⊆ R N is a bounded domain with a C 2  where C = {x ∈ W 1,p 0 (Z) : x(z) ≥ g(z) a.e on Z}, with g ∈ W 1,p (Z), g(z) ≤ 0 a.e. on Z, and u ∈ L q (Z), (1/ p + 1/q = 1, 1 < p < ∞), u(z) ∈ ∂ j(z,x(z)) a.e. on Z. Here, the potential function z → j(z,x) is only locally Lipschitz, not necessarily C 1 , and ∂ j(z,•) denotes the generalized (Clarke) subdifferential (see Section 2).In addition, λ 1 > 0 stands for the first eigenvalue of the negative p-Laplacian with Dirichlet boundary condition (denoted by (− p ,W 1,p 0 (Z))).Also, we study the following nonlinear elliptic problem at resonance with nonsmooth potential: Having as a starting point a solution of problem (1.1) when g ≡ 0, we show that problem (1.2) has at least one nontrivial positive solution.Moreover, by strengthening a little our hypotheses, we show that the solution is in fact strictly positive and smooth.
Problems similar to (1.1) were investigated by Szulkin [19] and more recently by Le [13,14] and Zhou and Huang [22].In [19], the problem under consideration is semilinear (i.e., p = 2), not in resonance, g ≡ 0, and the potential function is smooth (i.e., a C 1 function).His approach is variational.Le [13,14] considers nonlinear variational inequalities not in resonance, which though, are driven by general nonlinear differential operators, which include as a special case the p-Laplacian.The right-hand side nonlinearity is Carathéodory (thus the corresponding potential function is C 1 ) and it may also depend on the gradient of the unknown function (see Le [13]).His approach is based on the method of upper and lower solutions.Moreover, we mention that in Le [14] the interested reader can find a rich bibliography on the subject of variational inequalities.In Zhou and Huang [22], the problem is nonlinear, not in resonance, with smooth potential.The approach in that paper is based on the theory of nonlinear complementarity problems.
Problems like (1.2) were studied in the context of semilinear equations (i.e., p = 2) which are either nonresonant (see Alves and Miyagaki [1], Zhou [21]) or are near-resonance (see Mawhin and Schmitt [16] (ordinary differential equations) and Chiappinelli et al. [3] (elliptic equations)).For the near resonance problems, the authors obtain multiplicity results.For nonlinear problems driven by the p-Laplacian, the investigation is lagging behind and only recently Kyritsi and Papageorgiou [11] extended the aforementioned works on nearly resonant problems.
Unilateral problems with nonsmooth potential are also known in the literature as "variational-hemivariational inequalities" and arise in mechanics and engineering when one wants to consider more realistic nonmonotone and multivalued laws.We refer to the book of Naniewicz and Panagiotopoulos [17] and the paper of Goeleven et al. [7].We should mention that our work here generalizes in many respects the semilinear eigenvalue problems studied by Goeleven et al. [7].
Our approach is variational, based on the nonsmooth extension of the theory of Szulkin [19], due to Kourogenis et al. [9].For the convenience of the reader, in the next section we recall the basic definitions and facts from this theory, as well as relevant notions from convex analysis and the subdifferential theory for locally Lipschitz functions.For details, we refer to the books of Clarke [4] and Denkowski et al. [6].

Mathematical background
Let X be a Banach space and X * its topological dual.By •, • we denote the duality brackets for the pair (X,X * ).A function ϕ : X → R is said to be locally Lipschitz, if for all x ∈ X, we can find a neighborhood U of x and a constant k U > 0 such that |ϕ(y) − ϕ(u)| ≤ k U y − u , for all y,u ∈ U. From convex analysis, we know that if ψ : X → R = R ∪ {+∞} is convex, lower semicontinuous, and proper (i.e., ψ is not identically +∞), then ψ is locally Lipchitz in the interior of its effective domain domψ = {x ∈ X : ψ(x) < +∞}.In particular, then a continuous, convex function ψ : X → R is locally Lipschitz.In analogy to the directional derivative of a convex function, for a locally Lipschitz function ϕ : X → R, we define the generalized directional derivative of ϕ at x in the direction h ∈ X by It is easy to check that ϕ 0 (x;•) is sublinear continuous, so it is the support function of a nonempty, w * -compact, and convex set ∂ϕ(x) = {x * ∈ X * : x * ,h ≤ ϕ 0 (x;h) for all h ∈ X}.The multifunction ∂ϕ : X → 2 X * \ {∅} is called the generalized (or Clarke) subdifferential of ϕ.If ϕ is also convex, then the generalized subdifferential coincides with the subdifferential in the sense of convex analysis, given by are locally Lipschitz functions and λ ∈ R, then for all x ∈ X, we have Let Γ 0 (X) be the cone of convex, lower semicontinuous, proper functions.Let ϕ 1 : X → R be locally Lipschitz, ϕ 2 ∈ Γ 0 (X), and set ϕ = ϕ 1 + ϕ 2 .For such functions exists a nonsmooth critical point theory (see Kourogenis et al. [9]), which extends the theory of Szulkin [19], where We say that ϕ = ϕ 1 + ϕ 2 satisfies the generalized nonsmooth Palais-Smale condition (generalized nonsmooth PS-condition for short), if every sequence {x n } n≥1 ⊆ X such that {ϕ(x n )} n≥1 is bounded and there exists {ε n } n≥1 ⊆ R + , ε n ↓ 0 such that has a strongly convergent subsequence.Remark that if ϕ 2 ≡ 0, then using Szulkin [19, Lemma 3], we can find x * n ∈ X * , x * n ≤ 1, such that and we recover the nonsmooth PS-condition introduced by Chang [2] (see also Kourogenis and Papageorgiou [10] for extensions).
locally Lipschitz and ϕ 2 ∈ Γ 0 (X), ϕ satisfies the generalized nonsmooth PS-condition, and there exist r > 0 and x 0 ∈ X with x 0 > r such that then ϕ has a critical point x ∈ X with corresponding critical value c = ϕ(x) given by where Since our problems are at resonance, they involve the principal eigenvalue λ 1 of (− p , W 1,p 0 (Z)).So, very briefly, we recall what is known about λ 1 .For details, we refer to Denkowski et al. [5].So, on Z, we consider the following nonlinear eigenvalue problem: Every λ ∈ R for which (2.8) has a nontrivial solution is said to be an eigenvalue of (− p ,W 1,p 0 (Z)) and the nontrivial solution x ∈ W 1,p 0 (Z) is a corresponding eigenfunction.Using as a test function x ∈ W 1,p 0 (Z), we see that every eigenvalue λ is nonnegative.In fact, the first eigenvalue λ 1 is strictly positive, isolated, and simple (i.e., the corresponding eigenspace is one-dimensional).Moreover, there is a variational characterization of λ 1 > 0 via a Rayleigh quotient, This infimum is realized at the corresponding normalized eigenfunction u 1 .Note that in (2.9), we can replace x by |x|, and so we infer that u 1 does not change sign on Z.Moreover, from nonlinear regularity theory, we know that u 1 ∈ C 1 ( Z) and we can say that u 1 (z) > 0 for all z ∈ Z.

Generalized variational inequalities
In this section, we deal with problem (1.1).Our hypotheses on the nonsmooth potential j(z,x) are the following: (iii) for every M > 0, there exists α M ∈ L ∞ (Z) such that for almost all z ∈ Z, all |x| ≤ M, and all u ∈ ∂ j(z,x), we have |u| ≤ α M (z); (iv) lim x→+∞ (u/x p−1 ) = 0 uniformly for almost all z ∈ Z and all u ∈ ∂ j(z,x); and M. E. Filippakis and N. S. Papageorgiou 639 We assume that there exists a function Remark 3.1.Hypothesis H(j)(v) is an extension of the classical Landesman-Lazer condition.
. This is a consequence of the fact that ϕ 0 1 (x n ;•) is the support function of ∂ϕ 1 (x n ) and the latter is w-compact in W −1,q (Z) = W 1,p 0 (Z) * .For every n ≥ 1, we have Here, A : W 1,p 0 (Z) → W −1,q (Z) is the nonlinear operator defined by and We claim that {x n } n≥1 ⊆ W 1,p 0 (Z) is bounded.Suppose that this is not the case.By passing to a suitable subsequence if necessary, we may assume that x n → ∞.Set v n = x n / x n , n ≥ 1. Evidently, v n = 1, n ≥ 1, and so we may assume that By virtue of hypotheses H(j)(iii) and (iv), given ε > 0, we can find α ε ∈ L ∞ (Z) such that for almost all z ∈ Z, all x ≥ 0, and all u ∈ ∂ j(z,x), we have (3.7) Using the mean value theorem for locally Lipschitz functions (see Clarke [4, page 41] and Denkowski et al. [6, page 609]), for almost all z ∈ Z and all x ≥ 0, we have Passing to the limit as n → ∞ and using the weak lower semicontinuity of the norm functional, we obtain (see [4]).
If v = 0, then Dv n p → 0, and so v n → 0 in W 1,p 0 (Z), a contradiction to the fact that v n = 1 for all n ≥ 1.So, v = ±u 1 .Recall that x n (z) ≥ g(z) a.e. on Z, n ≥ 1.Hence, v n (z) ≥ g(z)/ x n a.e. on Z, and so, in the limit, we have v(z) ≥ 0 a.e on Z.Therefore, v = u 1 .Now we fix y = 0 ∈ C. From the choice of the sequence {x n } n≥1 ⊆ C, we have Then we have (see hypothesis H(j)(v) and recall that v n (z) ≥ 0 a.e. on Z).
Recall that v = u 1 and u 1 (z) > 0 for all z ∈ Z.It follows that x n (z) → +∞ a.e. on Z as n → ∞, and if by χ E we denote the characteristic function of a measurable set E ⊆ Z, we have Also, from the definition of the function G + (•) (see hypothesis H(j)(v)), given ε > 0, we can find M 2 > 0 such that for almost all z ∈ Z and all x > M 2 , we have (3.17) On the other hand, for almost all z ∈ Z and all x ∈ [0,M 2 ], we have Therefore, for almost all z ∈ Z and all x ∈ R + , we have Passing to the limit as n → ∞ in (3.15) and using Fatou's lemma (remark that (3.19) permits the use of Fatou's lemma), we obtain Z G + (z)u 1 (z)dz ≤ 0, (3.20) which contradicts hypothesis H(j)(v).Therefore, it follows that {x n } n≥1 ⊆ C is bounded in W 1,p 0 (Z), and so we may assume that x n w − → x in W 1,p 0 (Z) and x n → x in L p (Z), with x ∈ C. We have We fix y = x ∈ C. Hence we have Proof.Note that for all t ≥ 0, tu 1 ∈ C, and so Recall that given ε > 0, we can find M 2 = M 2 (ε) > 0 such that for all z ∈ Z\D, |D| N = 0 (by | • | N we denote the Lebesgue measure on R N ), and all x > M 2 , we have (3.26) For all z ∈ Z\D, all x > M 2 , and all u ∈ ∂ j(z,x), we have For all z ∈ Z\D, the function x → p j(z,x)/x p is locally Lipschitz on (M 2 ,+∞) and (see Clarke [4, page 48] and Denkowski et al. [6, page 612]).So, for all z ∈ Z\D, |D| N = 0, all x > M 2 , and all u ∈ ∂( j(z,x)/x p ), we have for almost all x ∈ M 2 ,+∞ . ( Consider w,v > M 2 , v > w, and integrate the last inequality on the interval [w,v].We obtain (3.31) We have seen in the proof of Proposition 3.2 that given ε > 0, we can find β ε ∈ L 1 (Z) + such that for all z ∈ Z\D, |D| N = 0, and all x ∈ R, we have x p = 0 since ε > 0 was arbitrary .
(3.32) So, returning to inequality (3.31) and passing to the limit as v → +∞, we obtain Suppose that the conclusion of the proposition was not true.This means that we can find M 3 > 0 and a sequence {t n } n≥1 ⊆ R + with t n → +∞, such that (3.34) Via Fatou's lemma and using (3.33), we obtain a contradiction to hypothesis H(j)(v).So, ϕ(tu 1 ) → −∞ as t → +∞.Proof.By virtue of Proposition 3.4, if we choose r > 0 small, then for all x ∈ W 1,p 0 (Z) with x = r, we have ϕ(x) ≥ ξ 2 > 0. (3.41) On the other hand, because of Proposition 3.3, we can find t > r such that ϕ(tu 1 ) ≤ ϕ(0) ≤ 0. These facts in conjunction with Proposition 3.2 permit the use of Theorem 2.1, which gives x ∈ C such that . Moreover, from the second inequality in (3.42), we have a.e on Z, and v * ,x − y ≥ 0 for all y ∈ C. It follows that x * , y − x ≥ 0 for all y ∈ C. Therefore, we have (3.46)

Positive solutions
In this section, we deal with problem (1.2).Having as our starting point the existence theorem of the previous section, we establish that problem (1.2) has a nontrivial positive solution.
.23) Recall that A is maximal monotone, thus generalized pseudomonotone (see Denkowski et al. [5, page 37]).So we have Dx n p p = A x n ,x n −→ A(x),x = Dx in L p (Z,R N ) and L p (Z,R N ) is uniformly convex, from the Kadec-Klee property, we have Dx n → Dx in L p (Z,R N ), hence x n → x in W 1,p 0 (Z).Proposition 3.3.If hypotheses H(j) hold, then ϕ(tu 1 ) → −∞ as t → +∞.