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Generalized eigenvalues of the (P, 2)-Laplacian under a parametric boundary condition

Part of: Elliptic equations and systems Representations of solutions Spectral theory and eigenvalue problems

Published online by Cambridge University Press:  18 December 2019

Jamil Abreu  and
Gustavo F. Madeira
Jamil Abreu
Affiliation:
Departamento de Matemática Aplicada, Universidade Federal do Espírito Santo, Rodovia BR101, Km 60, São MateusES, Brazil ([email protected])
Gustavo F. Madeira
Affiliation:
Departamento de Matemática, Universidade Federal de São Carlos – UFSCar, Rod. Washington Luís, Km 235 – São CarlosSP, Brazil ([email protected])
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Abstract

In this paper we study a general eigenvalue problem for the so called (p, 2)-Laplace operator on a smooth bounded domain Ω ⊂ ℝN under a nonlinear Steklov type boundary condition, namely

\[\left\{ \begin{aligned} -\Delta_pu-\Delta u & =\lambda a(x)u \quad {\rm in}\ \Omega,\\ (|\nabla u|^{p-2}+1)\dfrac{\partial u}{\partial\nu} & =\lambda b(x)u \quad {\rm on}\ \partial\Omega . \end{aligned} \right.\]
For positive weight functions a and b satisfying appropriate integrability and boundedness assumptions, 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.

Keywords

eigenvalue problemcontinuous family of eigenvalues(p, 2)-LaplacianSteklov boundary conditionboundary condition with eigenvalue parameter

MSC classification

Primary: 35D30: Weak solutions 35J60: Nonlinear elliptic equations
Secondary: 35P30: Nonlinear eigenvalue problems, nonlinear spectral theory
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 63 , Issue 1 , February 2020 , pp. 287 - 303
Copyright
Copyright © Edinburgh Mathematical Society 2019

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