Entire solutions of certain fourth order elliptic problems and related inequalities

Abstract

AbstractWe study distributional solutions of semilinear biharmonic equations of the typeΔ2u+f(u)=0 onℝN,{\Delta ^2}u + f(u) = 0\quad on\;{{\mathbb R} ^N},wherefis a continuous function satisfyingf(t)t≥c|t|q+1for allt∈ ℝ withc> 0 andq> 1. By using a new approach mainly based on careful choice of suitable weighted test functions and a new version of Hardy- Rellich inequalities, we prove several Liouville theorems independently of the dimensionNand on the sign of the solutions.