Existence and nonexistence of nontrivial solutions of some nonlinear fourth order elliptic equations

Abstract

In this paper, we are concerned with the existence and nonexistence of nontrivial solutions for nonlinear elliptic equations involving a biharmonic operator. Concerning the second order equations, a complementary result was obtained for the problem of interior, exterior and whole space. The main purpose of this paper is to discuss whether the complementary result mentioned above is still valid for the nonlinear fourth order equations. We introduce Kelvin type transformation for a biharmonic operator to convert an exterior problem to an interior problem. The existence results in case of super-critical exterior problem are shown by introducing a weighted version of Sobolev-Poincare type inequality, and the nonexistence results are shown by giving a Pohozaev-type identity for fourth order equations.