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In this work, we review and extend some well known results for the eigenvalues of the Dirichlet $p-$Laplace operator to a more general class of monotone quasilinear elliptic operators. As an application we obtain some homogenization results for nonlinear eigenvalues., 23 pages, Rev. UMA, to appear

In this note we give some remarks and improvements on our recent paper [5] about an optimization problem for the p-Laplace operator that were motivated by some discussion that we had with Prof. Cianchi. © 2009 Elsevier Ltd. All rights reserved. Fil:Del Pezzo, L.M. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina. Fil:Fernández Bonder, J. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina.

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.