Global Bifurcation Results for a Semilinear Biharmonic Equation on all of {I \hskip -2.8mm R}$^N$

N. M. Stavrakakis and N. Zographopoulos

Abstract: We prove existence of positive solutions for the semilinear problem $$ (-\Delta)^2 u = \lambda g(x)f(u), \qquad u(x) > 0 \ \ (x \in {\Bbb R}^N), \qquad {\textstyle \lim_{|x|\to+\infty}} u(x) = 0 $$ under the main hypothesis $N > 4$ and $g \in L^{N/4}({\Bbb R}^N)$. First, we employ classical spectral analysis for the existence of a simple positive principal eigenvalue 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

Classification (MSC2000): 35B32, 35B40, 35J40, 35J65, 35J70, 35H12

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