On the principal eigencurve of the $p$-Laplacian related to the Sobolev trace embedding
Volume 32 / 2005
Applicationes Mathematicae 32 (2005), 1-16
MSC: 35P30, 35J20, 35J60.
DOI: 10.4064/am32-1-1
Abstract
We prove that for any $\lambda \in {\Bbb R}$, there is an increasing sequence of eigenvalues $\mu_n(\lambda)$ for the nonlinear boundary value problem $$ \cases{ {\mit\Delta}_pu=|u|^{p-2}u &\textrm{in } {\mit\Omega} ,\cr |\nabla u|^{p-2}{\partial u}/{\partial \nu}=\lambda \varrho (x)|u|^{p-2}u + \mu|u|^{p-2}u &\textrm{on } \partial {\mit\Omega} ,\cr} $$ and we show that the first one $\mu_{1}(\lambda)$ is simple and isolated; we also prove some results about variations of the density $\varrho $ and the continuity with respect to the parameter $\lambda$.
