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Nonlinear Eigenvalue Problem for the p-Laplacian

Commun. Math. Anal.
Volume 20, Number 1 (2017), 69 - 82

Nonlinear Eigenvalue Problem for the p-Laplacian

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Abstract

This article is devoted to the study of the nonlinear eigenvalue problem
\begin{eqnarray*}
% \nonumber to remove numbering (before each equation)
-\Delta_{p} u &=& \lambda |u|^{p-2}u~~\text{in}~~\Omega,\\
|\nabla u|^{p-2}\frac{\partial u}{\partial \nu}&+&\beta |u|^{p-2}u=\lambda |u|^{p-2}u ~~\text{on }~~~\partial\Omega,
\end{eqnarray*}
where $\nu$ denotes the unit exterior normal, $10$. Using Ljusternik-Schnirelman theory, we prove the existence of a nondecreasing sequence of positive eigenvalues and the first eigenvalue is simple and isolated. Moreover, we will prove that the second eigenvalue coincides with the second variational eigenvalue obtained via the Ljusternik-Schnirelman theory.