Soluciones simétricas de algunos problemas elípticos

In this paper we study solutions to the Neumann problem

(I)         ∆u=  F(u)   in Ω,

   ∂u/∂n =  G(u)  on Ω,                                               

and the Dirichlet problema

     (II)    ∆u=F(u)   in  Ω,

              u=c        n  ∂Ω      

where Ω is a bounded domain in Rⁿ with a smooth boundary ∂ Ω  ∂/ ∂n is the derivative with respect to the outward normal n and c ϵ R. If  Ω is the unit ball and if either F(t) = f(t) and G(t) = g(t) or F(t) = /(t) . t and G(t) = 9(t) . t where f is a strictly increasing continuous function and  g is a strictly decreasing continuous function, we prove that solutions to problems (I) and (II) are radially symmetric about the origen. If Ω  is the unit ball and F is a continuous function that does not change sign, we prove that solutions of (II) are radially symmetric about the origen. If Ω ⊂ Rⁿ  is a symmetric bounded domain with respect to a hyperplane T and f ϵ C(Ω x R,R), g ϵC (∂Ω x R, R) are functions that satisfy the same monoton..icity properties in the second variable as before, then we prove that solutions are symmetric with respect to the hyperplane T. If F satisfies the same condition as in the first case and G ≡ 0, we prove that the only solutions of (I) are constant functions. Furthermore, we find a formula for solutions of (I) in the unitary ball that allow us to deduce some non-existence results. We find conditions on F and G in order for (I) to have no solutions in any bounded domain.