Abstract

Bifurcation from the Fu\v{c}ik-spectrum W. Reichel For $1<p<\infty$ nontrivial solutions of $${\rm div}(|\nabla u|^{p-2}\nabla u)+(\mu u^+-\nu u^-)|u|^{p-2} = 0 \mbox{ in } \Omega\subset\R^n $$ with homogeneous boundary conditions on a bounded domain $\Omega$ are considered. Here we write $u=u^+-u^-$, $u^+=\max\{u,0\}$. The case $p=2$ and $\mu=\nu=\lambda$ corresponds to a familiar linear eigenvalue problem $\Delta u +\lambda u = 0$. Properties of the Fu\v{c}ik-eigenvalues $(\mu,\nu)$ are described. In the presence of a higher-order perturbation $g(s)=o(|s|^{p-1})$ as $s\to 0$ the resulting problem $${\rm div}(|\nabla u|^{p-2}\nabla u)+(\mu u^+-\nu u^-)|u|^{p-2}+g(u) = 0 \mbox{ in } \Omega $$ shows bifurcation phenomena from the Fu\v{c}ik-eigenvalues, which resemble those of second-order linear Sturm-Liouville problems and their nonlinear perturbations. With increasing level of completeness we address the true PDE problem for a general bounded domain in $\R^n$, the radially symmetric problem on a ball in $\R^n$, the one-dimensional problem on an interval.

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