Classificação de soluções de algumas equações elípticas não lineraes

Abstract

In this work, we classify the solutions of the equation u + fue = 0 in R2 or R2 +. For this, we use basically the Moving Planes Method and and Moving Spheres Method. These methods ensure monotonicity and radial symmetry of the solution under certain conditions. The first method was used to study the case f 1 in R2 when RR2 eu is finite. The other was used to verify that the equation has no solution when f is a continuous function and radially symmetric, monotone in the region which has positive image and not constant. The latter method was also applied to the study of the problem ( u + eu = 0 em R2 +; @u @t = ceu=2 sobre @R2 +; for = 1; = 􀀀1 or = 0, modifying the conditions under the finiteness of RR2 + eu and R@R2 + eu=2. In most cases, when the equation has the solution, it was verified that the radially symmetrical. From this symmetry, we transform our Partial Differential Equations for Ordinary Differential Equations and we classify their solutions.