Global Bifurcation Results for a Semilinear Biharmonic Equation on all of ${\mathbb{R}}^N$

Abstract

We prove existence of positive solutions for the semilinear problem

under the main hypothesis $N>4$ and $g\in L^{N/4}({\mathbb{R}}^N)$. First, we employ classical spectral analysis for the existence of a simple positive principal cigenvalue for the linearized problem. Next, we prove the existence of a global continuum of positive solutions for the problem above, branching out from the first eigenvalue of the differential equation in the case that $f(u)=u$. This fact is achieved by applying standard local and global bifurcation theory. It was possible to carry out these methods by working between certain equivalent weighted and homogeneous Sobolev spaces.