SILVA, Lucas da.
Résumé:
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.
