Sobre existência de soluções para sistemas elípticos envolvendo operadores divergente com peso via teoria de pontos fixos em cones

Abstract

In this work, we will prove the existence of solutions for some elliptic systems involving divergent operators with weight, of the type:   −div(w1(x)Ñu) = w3(x) f (|x|,u, v), x ∈ B, −div(w2(x)Ñv) = w4(x)g(|x|,u, v), x ∈ B, u(x) = 0 = v(x), x ∈ ¶B, where B is the unitary ball from RN and w1,w2,w3,w4 are the weight functions. We studied one case where the operator associated with the system is linear and another case where the operator is nonlinear. We noted that each one of these cases has particular challenges despite both situations having similar general ideas. The existence of solutions is obtained by Fixed Points Theory in Cones, more specifically by a direct application of Krasnoselskii’s fixed point theorem.