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

• Autores: Julián Fernández Bonder, Julio D. Rossi
• Localización: Publicacions matematiques, ISSN 0214-1493, Vol. 46, Nº 1, 2002, págs. 221-235
• Idioma: inglés
• DOI: 10.5565/publmat_46102_12
• Títulos paralelos:
  • Un problema no lineal de autovalores con pesos indefinidos relacionado con la inmersión de trazas de Sobolev
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• Resumen

  • 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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