The \(\infty \)-eigenvalue problem with a sign-changing weight

  • Uriel Kaufmann
  • Julio D. Rossi
  • Joana TerraEmail author


Let \(\Omega \subset {\mathbb {R}}^{n}\) be a smooth bounded domain and \(m\in C(\overline{\Omega })\) be a sign-changing weight function. For \(1<p<\infty \), consider the eigenvalue problem
$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta _{p}u=\lambda m(x)|u|^{p-2}u &{}\quad \text {in}\;\; \Omega ,\\ u=0 &{}\quad \text {on}\;\; \partial \Omega , \end{array} \right. \end{aligned}$$
where \(\Delta _{p}u\) is the usual p-Laplacian. Our purpose in this article is to study the limit as \(p\rightarrow \infty \) for the eigenvalues \(\lambda _{k,p}\left( m\right) \) of the aforementioned problem. In addition, we describe the limit of some normalized associated eigenfunctions when \(k=1\).


Infinity Laplacian Eigenvalues Sign-changing weight Viscosity solutions 

Mathematics Subject Classification

35P15 35P30 35J60 



The research of UK was partially funded by Secyt-UNC 33620180100016CB (Argentina). The research of JDR was partially supported by CONICET grant PIP GI No 11220150100036CO (Argentina), by UBACyT grant 20020160100155BA (Argentina) and by MINECO MTM2015-70227-P (Spain). JT was partially supported by ANPCyT grant PICT 2016-1054 (Argentina) and by Secyt-UNC 33620180100016CB (Argentina).


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Authors and Affiliations

  1. 1.Depto. de Matemática FCEyNUniversidad de Buenos AiresBuenos AiresArgentina
  2. 2.FaMAFUniversidad Nacional de CórdobaCórdobaArgentina

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