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Eigenvalue Robin Problem for the p-Laplacian with Weight

Volume 3, Number 2 (2012), 60 - 74

Eigenvalue Robin Problem for the p-Laplacian with Weight

Communicated By: 
Alain Haraux
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Abstract

This article is devoted to the study of the eigenvalue Robin problem −Δpu = λm(x)|u|p−2u in Ω, |ru|p−2 ∂u ∂ν + β|u|p−2u = 0 on ∂Ω, where ν denotes the unit exterior normal, 1 < p < ∞ and Δpu = div(|ru|p−2ru) denotes the plaplacian. Ω RN is a bounded domain with smooth boundary where N ≥ 2 and β is non-negative constant. The weight function m 2 L∞(Ω) may change sign and has nontrivial positive part. Using Ljusternik-Schnirelman theory, we prove the existence of a nondecreasing sequence of positive eigenvalues and the first eigenvalue is simple, isolated and monotone with respect to the weight. Then we prove an nonexistence result related to the first eigenvalue and we end this article with the study of the second eigenvalue. We will prove that it coincides with the second variational eigenvalue obtained via the Ljusternik-Schnirelman theory.