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# Zeitschrift für Analysis und ihre Anwendungen

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**Volume 18, Issue 3, 1999, pp. 753–766**

**DOI: 10.4171/ZAA/910**

Published online: 1999-09-30

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

N.M. Stavrakakis and N. ZographopoulosWe prove existence of positive solutions for the semilinear problem $$(–\Delta)^2u = \lambda g(x)f(u), \ \ \ u(x) > 0 \ (x \in \mathbb R^N), \ \ \ \mathrm {lim}_{|x| \to +\infty} u(x) = 0$$ under the main hypothesis $N > 4$ 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.

*Keywords: *Biharmonic equations, nonlinear eigenvalue problems, local and global bifurcation theory, maximum principle, indefinite weights

Stavrakakis N.M., Zographopoulos N.: Global Bifurcation Results for a Semilinear Biharmonic Equation on all of $\mathbb R^N$. *Z. Anal. Anwend.* 18 (1999), 753-766. doi: 10.4171/ZAA/910