Generalized spectrum of the $(\mathbf{p},2)$-Laplacian under a parametric boundary condition

Abstract

In this paper we study an eigenvalue problem for the so called $(p,2)$-Laplace operator on a smooth bounded domain under a nonlinear Steklov type boundary condition, namely \begin{equation} \left\{ \begin{aligned} -Δ_pu-Δu & =λa(x)u \ \ \text{in}\ Ω,\\ (|\nabla u|^{p-2}+1)\dfrac{\partial u}{\partialν} & =λb(x)u \ \ \text{on}\ \partialΩ. \end{aligned} \right. \end{equation} Under suitable integrability and boundedness assumptions on the positive weight functions $a$ and $b$, we show that, for all $p>1$, the eigenvalue set consists of an isolated null eigenvalue plus a continuous family of eigenvalues located away from zero.

14 pages, title has been changed

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• Eigenvalues of the negative $(p,q)$-Laplacian under a Steklov-like boundary condition