JournalszaaVol. 18, No. 3pp. 753–766

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

  • N.M. Stavrakakis

    National Technical University of Athens, Greece
  • N. Zographopoulos

    National Technical University of Athens, Greece
Global Bifurcation Results for a Semilinear Biharmonic Equation on all of $\mathbb R^N$ cover
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Abstract

We prove existence of positive solutions for the semilinear problem

(Δ)2u=λg(x)f(u),   u(x)>0 (xRN),   limx+u(x)=0(–\Delta)^2u = \lambda g(x)f(u), \ \ \ u(x) > 0 \ (x \in \mathbb R^N), \ \ \ \mathrm {lim}_{|x| \to +\infty} u(x) = 0

under the main hypothesis N>4N > 4 and gLN/4(RN)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)=uf(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 RN\mathbb R^N. Z. Anal. Anwend. 18 (1999), no. 3, pp. 753–766

DOI 10.4171/ZAA/910