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

### N.M. Stavrakakis

National Technical University of Athens, Greece### N. Zographopoulos

National Technical University of Athens, Greece

## 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.

## Cite this article

N.M. Stavrakakis, N. Zographopoulos, Global Bifurcation Results for a Semilinear Biharmonic Equation on all of $\mathbb R^N$. Z. Anal. Anwend. 18 (1999), no. 3, pp. 753–766

DOI 10.4171/ZAA/910