Generalized eigenvalues of the (P, 2)-Laplacian under a parametric boundary condition
Jamil Abreu, Gustavo Ferron Madeira · Proceedings of the Edinburgh Mathematical Society · 2019
Abstract In this paper we study a general eigenvalue problem for the so called (p, 2)-Laplace operator on a smooth bounded domain Ω ⊂ ℝNunder a nonlinear Steklov type boundary condition, namely \[\left\{ \begin{aligned} -\Delta_pu-\Delta u & =\lambda a(x)u \quad {\rm in}\ \Omega,\\ (| abla u|^{p-2}+1)\dfrac{\partial u}{\partial u} & =\lambda b(x)u \quad {\rm on}\ \partial\Omega . \end{aligned} \right.\] For positive weight functionsaandbsatisfying appropriate integrability and boundedness assumptions, we show that, for allp>1, the eigenvalue set consists of an isolated null eigenvalue plus a continuous family of eigenvalues located away from zero.