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On the generalization of the Courant nodal domain theorem. (English) Zbl 1163.35449

This paper concerns with the nodal properties of the eigenfunctions (Fučik eigenfunctions) of $-{{\Delta }}_{p}$, where $p>1$, ${{\Delta }}_{p}u:=\nabla ·{\left(|\nabla u|}^{p-2}\nabla u\right)$ is the $p$-Laplacian. More precisely, the authors study the properties of the nonlinear eigenvalue problem

$-{{\Delta }}_{p}u=\lambda {|u|}^{p-2}u\phantom{\rule{1.em}{0ex}}\text{in}\phantom{\rule{4.pt}{0ex}}{\Omega },\phantom{\rule{2.em}{0ex}}u=0\phantom{\rule{1.em}{0ex}}\text{on}\phantom{\rule{4.pt}{0ex}}\partial {\Omega },$

and its more general version

$-{{\Delta }}_{p}u={\alpha |u|}^{p-2}{u}^{+}-\beta {|u|}^{p-2}{u}^{-}\phantom{\rule{1.em}{0ex}}\text{in}\phantom{\rule{4.pt}{0ex}}{\Omega },\phantom{\rule{2.em}{0ex}}u=0\phantom{\rule{1.em}{0ex}}\text{on}\phantom{\rule{4.pt}{0ex}}\partial {\Omega },$

where ${\Omega }$ is a bounded domain in ${ℝ}^{N}$ with smooth boundary $\partial {\Omega }$, and $\alpha ,\beta ,\lambda$ are real spectral parameters. The authors prove that, if ${u}_{{\lambda }_{n}}$ is an eigenfunction associated with the $n$th variational eigenvalue, ${\lambda }_{n}$, then ${u}_{{\lambda }_{n}}$ has at most $2n-2$ nodal domains. Moreover, if ${u}_{{\lambda }_{n}}$ has $n+k$ nodal domains then there is another eigenfunction with at most $n-k$ nodal domains.

##### MSC:
 35P30 Nonlinear eigenvalue problems for PD operators; nonlinear spectral theory 47J10 Nonlinear spectral theory, nonlinear eigenvalue problems 58E05 Abstract critical point theory