EXISTENCE RESULTS FOR GENERAL INEQUALITY PROBLEMS WITH CONSTRAINTS

This paper is concerned with existence results for inequality problems of type F0(u;v) + Ψ′(u;v) ≥ 0, for all v ∈ X , where X is a Banach space, F : X → R is locally Lipschitz, and Ψ : X → (−∞+∞] is proper, convex, and lower semicontinuous. Here F0 stands for the generalized directional derivative of F and Ψ′ denotes the directional derivative of Ψ. The applications we consider focus on the variational-hemivariational inequalities involving the p-Laplacian operator.


Introduction
The paper deals with nonlinear inequality problems of type where F 0 stands for the generalized directional derivative of a locally Lipschitz functional F (in the sense of Clarke [5]), h is a convex, lower semicontinuous (in short, l.s.c.), and proper function, and C is a nonempty, closed, and convex subset of a Banach space X.It is clear that in problem (1.1) we can put h + I C in place of h, where I C denotes the indicator function of the set C, to give the formulation with v arbitrary in X.However, we keep the statement (1.1) for allowing various possible choices separately on the data h and C.
The type of problem stated in (1.1) fits in the framework of the nonsmooth critical point theory developed by Motreanu and Panagiotopoulos [9], which is constructed for the nonsmooth functionals having the form (1.1) means, in fact, a critical point of the associated nonsmooth functional (1.2) with Ψ = h + I C .
In particular, this contains the differential inclusion problem which we considered in our previous paper [8].
The rest of the paper is organized as follows.In Section 2, we briefly recall several elements of nonsmooth critical point theory developed by Motreanu and Panagiotopoulos [9].In Section 3, we study some general inequality problems in relation with the nonsmooth critical point theory.Section 4 presents applications for different discontinuous boundary value problems with p-Laplacian.

Notions and preliminary results
Let X be a real Banach space and X * its dual.The generalized directional derivative of a locally Lipschitz function F : X → R at u ∈ X in the direction v ∈ X is defined by The generalized gradient (in the sense of Clarke [5]) of F at u ∈ X is defined to be the subset of X * given by where •, • stands for the duality pairing between X * and X.Let Ψ : X → (−∞,+∞] be a proper (i.e., D(Ψ) := {u ∈ X : Ψ(u) < +∞} = ∅), convex, and l.s.c.function and let F : X → R be locally Lipschitz.

George Dincȃ et al. 603
Definition 2.1 Motreanu and Panagiotopoulos [9].An element u ∈ X is called critical point of the functional Φ if this inequality holds Definition 2.2 Motreanu and Panagiotopoulos [9].The functional Φ is said to satisfy the Palais-Smale condition if every sequence {u n } ⊂ X for which Φ(u n ) is bounded and for a sequence {ε n } ⊂ R + with ε n → 0, contains a strongly convergent subsequence in X.
(ii) If Φ satisfies the Palais-Smale condition and there exist a number ρ > 0 and a point e ∈ X with e > ρ such that then Φ has a nontrivial critical point.

Critical points as solutions of inequality problems
Throughout this section, (X, • X ) is a real reflexive Banach space, compactly embedded in the real Banach space (Z, • Z ).Let Ᏺ : Z → R be a locally Lipschitz function and let Ψ : X → (−∞,+∞] be convex, l.s.c., and proper.We consider the inequality problem: where (Ᏺ| X ) 0 denotes the generalized directional derivative of the restriction Ᏺ| X while Ψ (u;v) is the directional derivative of the convex function Ψ at u in the direction v (which is known to exist).Note that if the Gâteaux differential dΨ(u) of Ψ at u ∈ D(Ψ) exists, then dΨ(u),v = Ψ (u;v), for all v ∈ X.
Proposition 3.1.Each solution of problem (3.1) solves the problem: If, in addition to our assumptions, X is densely embedded in Z, then problems (3.1) and (3.2) are equivalent.
Proof.For u,v ∈ X, the inequality below holds This becomes an equality if X is continuously and densely embedded in Z (see [5, pages 46-47] and [9, pages 10-12]).
Our approach for studying problem (3.1) is variational and relies on the use of the functional which is clearly of the form required in the previous section with F = Ᏺ| X .The next result points out the relationship between the critical points of the functional Φ in (3.4) and the solutions of problem (3.1).
then u is a solution of problem (3.1).
If Ψ is continuous at u, then a standard result of convex analysis (see Barbu and Precupanu [3, page 106]) allows to write Using the definition of the subdifferential ∂Ψ(u), we obtain (3.5).

George Dincȃ et al. 605
Remark 3.3.In view of Proposition 3.2(i), each result stating the existence of critical points for Φ in (3.4) asserts a fortiori existence of solutions to problem (3.1).
then Φ has a critical point.
Proof.The compact embedding of X into Z implies that Ᏺ| X is weakly continuous.We infer that Φ is sequentially weakly l.s.c. on X.Then, by standard theory, Φ is bounded from below and attains its infimum at some u ∈ X.From Theorem 2.3(i), u is a critical point of Φ.
Towards the application of Theorem 2.3(ii) to the functional Φ, we have to know when Φ satisfies the Palais-Smale condition.The following lemma provides a useful sufficient condition that improves the usual results based on the celebrated hypothesis (p 5 ) in [2] or (p 4 ) in [10].
Lemma 3.5.Assume, in addition, that Ψ and Ᏺ, entering the expression of Φ in (3.4), satisfy the following hypotheses: Then the functional Φ satisfies the Palais-Smale condition in the sense of Definition 2.2.
Proof.Let {u n } be a sequence in X for which there is a constant M > 0 with and inequality (2.4) holds for F = Ᏺ| X and a sequence ε n → 0 + .By (3.12), each u n is in D(Ψ).For t > 0, set v = (1 + t)u n in (2.4) with F = Ᏺ| X .Dividing by t and then letting t 0, one obtains that Inequalities (3.12) and (3.13) ensure that for n sufficiently large, one has Using (3.9) and (3.10), we find that As {u n } is bounded in X, we infer from (3.17) and the upper semicontinuity of We can now state the following result.

George Dincȃ et al. 607 (H3)
There exists an element u ∈ D(Ψ) such that Then Φ has a nontrivial critical point u ∈ X.In particular, problem (3.1) has a nontrivial solution.
Proof.We apply Theorem 2.3(ii) to the functional Φ in (3.4).Lemma 3.5 guarantees that Φ satisfies the Palais-Smale condition.It remains to check that Φ verifies condition (2.5) with e X > ρ.To this end, we prove that one can choose e = tu (with u entering (H3)) if t > 0 is sufficiently large.First, note that u = 0. Indeed, from (3.9), (3.10), and (3.21), we have which leads to a contradiction with (3.22) if u = 0. We observe that, due to the fact that u ∈ D(Ψ) and since D(Ψ) is a cone, the convex function s → Ψ(su) is locally Lipschitz on (0,+∞).A straightforward computation shows that where the notation ∂ s stands for the generalized gradient with respect to s.For an arbitrary t > 1, Lebourg's mean value theorem yields some where ξ ∈ ∂ s (s −1/α Φ(su))| s=τ .This implies Then, taking into account the convexity of s → Ψ(su), the regularity property of a convex function (see Clarke [5, pages 39-40]) and relations (3.9) and (3.10), we get that (3.28)By (3.21) and because τ > 1, we derive that Now, by means of (3.30), we can choose t > 0 sufficiently large to satisfy for ρ > 0 entering (H4).If we compare (3.23) and (3.31), it is seen that the requirement in (2.5) is achieved for e = tu.Theorem 2.3(ii) assures that Φ in (3.4) has a nontrivial critical point u ∈ X.Furthermore, Remark 3.3 shows that u is a (nontrivial) solution of problem (3.1).The proof of Theorem 3.7 is thus complete.
In the final part of this section, we are concerned with the case when where C is a nonempty, closed, and convex subset of X, I C denotes the indicator function of C, and ϕ : X → R is a convex, Gâteaux differentiable functional.Note that Ψ C is convex, l.s.c., and proper and D(Ψ C ) = C. Therefore, the functional with Ᏺ as at the beginning of this section, has the form required in (3.4).
Consider the following problem of variational-hemivariational inequality type: If C is a nonempty, closed, and convex cone, then each solution of problem (3.34) solves also the problem: Proof.It is a direct consequence of Theorem 3.4 and Proposition 3.9.
Remark 3.12.It is worth pointing out that if we take C = X, then problem (3.34) becomes

Applications to nonsmooth boundary value problems
In order to illustrate how the abstract results of Section 3 can be applied, we consider a concrete problem of type (3.34).To this end, let Ω be a bounded domain in R N , N ≥ 1, with Lipschitz-continuous boundary Γ = ∂Ω and let ω ⊂ Ω be a measurable set.Given p ∈ (1,∞), the Sobolev space W 1,p (Ω) is endowed with its usual norm (see [1, page 44]).We denote In the sequel, W will stand for any of the above (closed) subspaces W 0 , W 1 , and W 2 of W 1,p (Ω).By the Poincaré-Wirtinger inequality, the functional is a norm on W, equivalent to the induced norm from W 1,p (Ω).The dual space W * is considered endowed with the dual norm of • 1,p .Now, we define the p-Laplacian operator Arguments similar to those in [7] show that the convex functional ϕ : W → R defined by George Dincȃ et al. 611 is continuously differentiable on W and its differential is −∆ p , that is, Moreover, as dϕ is the duality mapping on W, corresponding to the gauge function t → t p−1 and because W is uniformly convex, dϕ satisfies condition (S + ) (see Remark 3.6).
If p * stands for the Sobolev critical exponent, that is, then, for any fixed q ∈ (1, p * ), by the Rellich-Kondrachov theorem, the embedding W L q (Ω) is compact (the space L q (Ω) is understood with its usual norm • 0,q ).The results in Section 3 will be applied by taking X = W, Z = L q (Ω), and ϕ defined in (4.4).
Further, to complete the setting, let a function g : Ω × R → R be measurable and satisfy the growth condition where c 1 ,c 2 ≥ 0 are constants.For a.e.x ∈ Ω and all s ∈ R, we put The following condition will be invoked below: g and g are N-measurable (4.9) (recall that a function h : Taking into account (4.11), we define the functional Ᏻ : L q (Ω) → R by putting It is known (see, e.g., Chang [4]) that Ᏻ is Lipschitz continuous on the bounded subsets of L q (Ω).At this stage, we introduce the closed convex cone K in W: and we formulate the problem: with ϕ in (4.4),I K the indicator function of the cone K in (4.13), has the form required in (3.33) and (3.32).We also need to invoke the following constant, depending on the cone K in the Banach space W: For u ∈ K, we put   Then, from (4.15), it follows that .25) By (4.17), we infer Proof.We will apply Theorem 3.11.Without loss of generality, we may suppose in (4.7) that q ∈ (p, p * ).For u ∈ K (see (4.13)), the sets Ω − and Ω + will be considered as being defined by (4.21), and recall that Ω − ⊂ Ω \ ω.