Periodic solutions of Euler–Lagrange equations in an anisotropic Orlicz–Sobolev space setting

Fernando D Mazzone,Sonia Acinas

doi:10.33044/revuma.v60n2a03

Abstract

We consider the problem of finding periodic solutions of certain Euler–Lagrange equations which include, among others, equations involving the p-Laplace operator and, more generally, the pp, qq-Laplace operator. We employ the direct method of the calculus of variations in the framework of anisotropic Orlicz–Sobolev spaces. These spaces appear to be useful in formulating a unified theory of existence of solutions for such a problem.

Highlights

Let Φ : Rd Ñ r0, `8q be a differentiable, convex function such that Φp0q “ 0, Φpyq ą 0 if y ‰ 0, Φpyq “ Φpyq, and lim where | ̈ | denotes the euclidean norm on Rd
For T ą 0, we assume that F : r0, T s Rd Ñ Rd (F “ F pt, xq) is a differentiable function with respect to x for a.e. t P r0, T s
We suggest the article [27] for definitions and main results on anisotropic Orlicz spaces

Summary

Introduction

Let Φ : Rd Ñ r0, `8q be a differentiable, convex function such that Φp0q “ 0, Φpyq ą 0 if y ‰ 0, Φpyq “ Φpyq, and lim. The goal of this paper is to obtain existence of solutions for the following problem:. We suggest the article [27] for definitions and main results on anisotropic Orlicz spaces These spaces allow us to unify and extend previous results on existence of solutions for systems like (PΦ). Our results still improve some results on pp, p2q-Laplacian systems since we obtain existence of solutions for them under less restrictive conditions. The authors dealt with the differentiability of such action integrals assuming, for the sake of simplicity, that the convex function Φ and its complementary function Φ‹ satisfy the ∆2-condition, which implies that. Since we are interested in considering functions that grow faster than those that satisfy the ∆2-condition, in the present paper we develop another proof for differentiability of action integrals. This theorem unifies and extends several results obtained in the previously cited bibliography

Anisotropic Orlicz and Orlicz–Sobolev spaces

Existence of minimizers

Regularity of minimizers and solutions of Euler–Lagrange equations