Liouville theorems for stable radial solutions for the biharmonic operator

Abstract

This article is concerned with the non-existence of stable solutions for a fourth-order semilinear elliptic equation Δ²u=f(u) in R^N, where f is a smooth nonlinearity. We establish the non-existence of stable radial solutions which verify decay conditions at infinity. Our Liouville-type results do not depend on the specific nonlinearity f. Moreover, in low dimensions and under no radial symmetry assumption, we prove Liouville-type results when f is increasing.