Abstract
AbstractThe purpose of this paper is to study the existence of weak solutions for some classes of hemivariational problems in the Euclidean space ℝd(d≥ 3). These hemivariational inequalities have a variational structure and, thanks to this, we are able to find a non-trivial weak solution for them by using variational methods and a non-smooth version of the Palais principle of symmetric criticality for locally Lipschitz continuous functionals, due to Krawcewicz and Marzantowicz. The main tools in our approach are based on appropriate theoretical arguments on suitable subgroups of the orthogonal groupO(d) and their actions on the Sobolev spaceH1(ℝd). Moreover, under an additional hypotheses on the dimensiondand in the presence of symmetry on the nonlinear datum, the existence of multiple pairs of sign-changing solutions with different symmetries structure has been proved. In connection to classical Schrödinger equations a concrete and meaningful example of an application is presented.
Highlights
The purpose of this paper is to study the existence of weak solutions for some classes of hemivariational problems in the Euclidean space
The proof of the above result is based on variational method in the nonsmooth setting
In order to recover compactness, the first task is to construct certain subspaces of H 1 (Rd ) containing invariant functions under special actions defined by means of carefully chosen subgroups of the orthogonal group O(d)
Summary
Let (X, k · kX ) be a real Banach space. We denote by X ∗ the dual space of X, whereas h·, ·i denotes the duality pairing between X ∗ and X. Let X be a reflexive real Banach space and let Φ, Ψ : X → R be locally Lipschitz continuous functionals such that Φ is sequentially weakly lower semicontinuous and coercive. An action of a compact Lie group G on the Banach space (X, k · kX ) is a continuous map. Let X be a Banach space, let G be a compact topological group acting linearly and isometrically on X, and J : X → R a locally Lipschitz, G-invariant functional. The functional is locally Lipschitz continuous and its critical points solve (Sλ ). Jλ is the sum of the C 1 (H 1 (Rd )) functional u 7→ kuk2 /2 and of the locally Lipschitz continuous functional Ψ, see Lemma 6.
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