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

The $$\infty $$ ∞ -eigenvalue problem with a sign-changing weight