Existência de soluções positivas para o p-Laplaciano com dependência do gradiente

Abstract

The main aim of this work is to prove the existence of positive solutions for Dirichlet problems involving the p-Laplacian operator and nonlinearities that depend on the gradient of the solution. We will consider the problem \Delta _{p}u = \omega (x)f(u, | \nabla u|) in a smooth bounded domain of R^{N}, where \omega is a weight function and f(u, | \nabla u|) is a nonlinearity. No asymptotic behavior is assumed on f. Such hypotheses will be replaced by appropriate conditions in a neighborhood of the first p-Laplacian eigenvalue. If \Omega is a radial domain, the existence of positive solutions will be obtained by applying the Schauder. Fixed Point Theorem. In the general case, we will apply the sub- and supersolution method. A subsolution will be obtained from the radial solution in the subdomain B_{} \subset \Omega and a supersolution will be obtained as a multiple of a solution of a linear problem in a domain \Omega_{2} supset \Omega. We will study the choice of the domain 2 and our results will be applied to guarantee the existence of positive solutions for the problem \Delta _{p}u = \lambda u(x) ^{q1}(1 + |\nabla u(x)|^{p}), with Dirichlet boundary condition, in smooth and bounded domain of R^{N}.