Existência de soluções positivas para uma classe de problemas elípticos em R^N

Abstract

In this work, we study the existence of positive solutions for the following class of problems:    −ε p∆pu + V (x)u p−1 = H(u), em R N , u > 0 em R N , u ∈ W1,p(R N ), (Pε) where ε > 0 is a positive parameter, 2 ≤ p < N, ∆pu = div(|∇u| p−2∇u) denotes the p-Laplacian operator, H : R → R is a continuous function satisfying some conditions and V : R → R is a function of class C 2 which belongs to two classes of potentials. Our study is divided in two parts: firstly we show the same results obtained by Alves (2015) for p ≥ 2, establishing the existence of positive solution for the (P ) problem when H has subcritical growth; in the second we show the existence of positive solution considering H with critical growth. The main tools used are the Variational Methods, Mountain Pass Theorem, Lions’ Concentration-Compactness Principle and del Pino and Felmer’s Penalization Method.