Quasilinear problems on non-reflexive Orlicz-Sobolev spaces.

Abstract

The goal of this paper is to study the existence of solutions for some classes of elliptic PDEs involving the -Laplacian operator, . Firstly, in order to generalize the results obtained in the paper [10], we present the study of two quasilinear Schrödinger equations with potential vanishing at inőnity and the N-function ˜ (Complementary of function ) may not satisfy the 2-condition. Here we present new compact embeddings in RN that are commonly known as Hardy-Type inequalities. These inequalities, associated with a Mountain Pass Theorem without the Palais-Smale condition for Gateaux-diferentiable energy functionals (Ghoussoub-Preiss Mountain Pass Theorem), yield solutions for the classes of problems initially studied. It is worth noting that in one of the classes, we assume that the nonlinearity of the problem is a non-local type with a Stein-Weiss convolution term. The regularity of the solutions was obtained using the regularity results due to Lieberman [24]. In a second part of this thesis, we study the existence of solutions for two classes of quasilinear systems driven by the operators 1 ( 1-Laplacian) and 2 ( 2-Laplacian) where the N-functions 1 and 2 or ˜ 1 and ˜ 2 may not satisfy the 2- condition. In the őrst class, we relax the 2-condition of the functions i(i = 1, 2) and present a deőnition for the well-known Ambrosetti-Rabinowitz condition for nonlinearity. In this class we base the results on a Rabinowitz saddle point theorem without the Palais-Smale condition for diferentiable Fréchet functionals combining with properties of the weak topology∗. In the second class, we relax the 2-conditions of the N-functions ˜ i(i = 1, 2) and assume that the nonlinearity has supercritical growth. Here, we use a link theorem without the Palais-Smale condition for locally Lipschitz functionals and combine it with a concentration-compactness lemma for non-reŕexive Orlicz-Sobolev space to guarantee the existence of solutions for this class.