Existence and infinitely many solutions for an abstract class of hemivariational inequalities
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A general method is given in order to guarantee at least one nontrivial solution, as well as infinitely many radially symmetric solutions, for an abstract class of hemivariational inequalities. This abstract class contains some special cases studied by many authors. We remark that, differently from the classical literature, in the proofs we use the Cerami compactness condition and the principle of symmetric criticality for locally Lipschitz functions. 1. Introduction Let (H, ·, ·H ) be a real separable Hilbert space and we suppose that the inclusions H Ll (RN ) are continuous with the embedding constants C(l), where l ∈ [2, p0 ] (2 < p0 < 2 = 2N/(N − 2)). We denote by · H the norm induced on H by the inner product ·, ·H and by · l the norm of Ll (RN ). Let f : RN × R → R be a continuous function. Several studies have appeared dealing with the existence and multiplicity of nonzero solutions u ∈ H of the equation (E) u,v H =
RN
f x,u(x) v(x)dx,
∀v ∈ H.
(1.1)
Existence and multiplicity results in some special cases of (E) were studied in many papers, see for instance Strauss [20], Bartsch and Willem [3, 4], Bartsch and Wang [2], and in the monographs of Kavian [8], Struwe [21], and Willem [22]. Now, let f : RN × R → R be a measurable function, and consider a real number 2 < p < p0 , and we suppose that the function f satisfies the growth condition (f1 ) | f (x,s)| ≤ c(|s| + |s| p−1 ) for a.e. x ∈ RN , for all s ∈ R, where c > 0 is a positive constant. In what follows, we use only that the functions h1 (u) = c|u| and h2 (u) = c|u| p−1 are convex, increasing, and h1 (0) = h2 (0) = 0. Let F : RN × R → R be the function defined by F(x,u) =
u 0
f (x,s)ds,
for a.e. x ∈ RN , ∀s ∈ R.
Copyright © 2005 Hindawi Publishing Corporation Journal of Inequalities and Applications 2005:2 (2005) 89–105 DOI: 10.1155/JIA.2005.89
(1.2)
90
Hemivariational inequalities For a.e. x ∈ RN , we have F(x,u) − F(x,v) ≤ c1 |u − v | h1 |u| + h1 |v | + h2 |u| + h2 |v | ,
(1.3)
where c1 is a constant which does not depend on u and v. Therefore, the function F(x, ·) is locally Lipschitz and we can define the (partial) Clarke derivative of it, that is, F(x, y + tw) − F(x, y) , t y →u, t →0+
F20 (x,u;w) = limsup
(1.4)
for a.e. x ∈ RN and for all u,w ∈ R. Now we formulate the following hemivariational inequality problem. Problem 1.1. Find u ∈ H such that (P) u,v H +
RN
F20 x,u(x); −v(x) ≥ 0,
∀v ∈ H.
(1.5)
To study the existence of solutions of problem (P), we introduce the functional Ψ : H → R defined by Ψ(u) = (1/2)u2H − Φ(u), where Φ(u) = RN F(x,u(x))dx. We will see that the critical points (in the sense of Chang) of the functional Ψ are the solutions of problem (P). Therefore, it is enough to study the existence of the critical points of the functional Ψ. Such problems appear in the nonsmooth mechanics, see the books of Panagiotopoulos [17, 18], Motreanu and Panagiotopoulos [14]. The study of problem (P) is motivated by these books and by the aforementioned papers. We e
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