Revistes Catalanes amb Accés Obert (RACO)

A nonlinear eigenvalue problem with indefinite weights related to the Sobolev trace embedding

J. Fernández Bonder, J. D. Rossi


In this paper we study the Sobolev trace embedding $W^{1,p}(\Omega)\hookrightarrow L^p_V(\partial \Omega)$, where $V$ is an indefinite weight. This embedding leads to a nonlinear eigenvalue problem where the eigenvalue appears at the (nonlinear) boundary condition. We prove that there exists a sequence of variational eigenvalues $\lambda_k\nearrow +\infty$ and then show that the first eigenvalue is isolated, simple and monotone with respect to the weight. Then we prove a nonexistence result related to the first eigenvalue and we end this article with the study of the second eigenvalue proving that it coincides with the second variational eigenvalue.

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