Equações elípticas com dependência não linear do gradiente

Abstract

In this thesis we study the existence of solutions for elliptic equations with nonlinear dependence on the gradient of the solution. More precisely, we guarantee the existenceof solution for the following classes of problems.(P1)_2u + q_u + _(x)u = f(x; u;ru;_u) in u(x) = 0; _u(x) = 0 on @;where _ RN;N _ 5, is a bounded smooth domain.(P2) uiv + qu00 + _(x)u = f(x; u; u0; u00) x 2 R:(P3)8<_u = h(x; u) + g(x;ru) in u > 0 in u = 0 on @;where is a bounded, smooth domain in RN;N _ 3, the function h has sublinear and singular terms and g is bounded from above by a convection term of the type jruj_ with _ > 0.(P4)Z uds + b jruj_ds_u = f(x; u;ru) em u(x) > 0 em u(x) = 0 sobre @;where _; > 0, a; b 2 R, _ RN, N _ 3, is a bounded, smooth domain andf : _ R _ RN ! R, H : R ! R are nonnegative continuous functions.In this work, the main techniques used to study these problems were: variational Techniques, Galerkin's method and Krasnoselskii's _xed point Theorem.Keywords: elliptic equations, nonlinear dependence on the gradient, variational techniques, Galerkin's Method, _xed point Theorem, bootstrap.