Generalized spectrum of the $\boldsymbol{(p,2)}$-Laplacian under boundary condition with eigenvalue parameter
Jamil Abreu, Gustavo Ferron Madeira · arXiv (Cornell University) · 2015
In this paper we present a preliminary study on an general 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} -\Delta_pu-\Delta u & =\lambda a(x)u \text{in} \Omega, (| abla u|^{p-2}+1)\dfrac{\partial u}{\partial u} & =\lambda b(x)u \text{on} \partial\Omega . \end{aligned} \right. \end{equation} For positive essentially bounded weights $a$ 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.