EIGENVALUES OF THE RADIALLY SYMMETRIC $p$-LAPLACIAN IN $\mathbb{R^n}$

Abstract

For the $p$-Laplacian $\Delta_p v \,{=}\, {\rm div}\:(|\nabla v|^{p-2}\nabla v)$, $p\,{>}\,1$, the eigenvalue problem $-\Delta_p v + q(|x|)|v|^{p-2}v \,{=}\, \lambda |v|^{p-2}v$ in $\R^n$ is considered under the assumption of radial symmetry. For a first class of potentials $q(r)\,{\to}\,\infty$ as $r\,{\to}\,\infty$ at a sufficiently fast rate, the existence of a sequence of eigenvalues $\lambda_k\,{\to}\,\infty$ if $k\,{\to}\,\infty$ is shown with eigenfunctions belonging to $L^p(\R^n)$. In the case $p\,{=}\,2$, this corresponds to Weyl's limit point theory. For a second class of power-like potentials $q(r)\,{\to}\,{-}\infty$ as $r\,{\to}\,\infty$ at a sufficiently fast rate, it is shown that, under an additional boundary condition at $r\,{=}\,\infty$, which generalizes the Lagrange bracket, there exists a doubly infinite sequence of eigenvalues $\lambda_k$ with $\lambda_k \,{\to}\,\pm \infty$ if $k\,{\to}\,\pm\infty$. In this case, every solution of the initial value problem belongs to $L^p(\R^n)$. For $p\,{=}\,2$, this situation corresponds to Weyl's limit circle theory.