Existência e multiplicidade de soluções positivas para problemas elípticos semilineares.

Abstract

In this dissertation, we present results of existence and multiplicity of positive solutions for the following problems: 8< :􀀀�Dv+lv = f (z)vp􀀀�1+h(z)vq􀀀�1, in RN v 2 H1(RN), (El) where 1 q<2< p<2 = 2N N􀀀�2 for N 3 and the functions f and g satisfy some conditions, and 8>>><>>>: 􀀀�Du = lg(z)jujp􀀀�2u+ a a+b f (z)juja􀀀�2ujvjb, in W 􀀀�Dv = μh(z)jvjp􀀀�2v+ b a+b f (z)jujajvjb􀀀�2v, in W u = v = 0, on ¶W, (El;μ) where a > 1, b > 1 and 2 < p < a+b = 2 = 2N N􀀀�2 , N > 4, l;μ > 0, 0 2 W RN is a bounded domain with smooth boundary ¶W and f ;g;h : W!R satisfy some conditions. Among the main tools used are the Ekeland’s Variational Principle, Pierre-Louis Lions Lemma, Lagrange Multipliers Theorem and properties involving Nehari Manifold.