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Isolation and simplicity for the first eigenvalue of the \(p\)-Laplacian with a nonlinear boundary condition. (English) Zbl 1065.35215

Summary: We prove the simplicity and isolation of the first eigenvalue for the problem \(\Delta_p u= | u|^{p-2}u\) in a bounded smooth domain \(\Omega \supset {\mathbb R}^{N}\), with a nonlinear boundary condition given by \(| \nabla u|^{p-2}\partial u/\partial v=\lambda | u|^{p-2}u\) on the boundary of the domain.

MSC:

35P05 General topics in linear spectral theory for PDEs
35J60 Nonlinear elliptic equations
35J25 Boundary value problems for second-order elliptic equations
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