Weighted Sobolev Spaces and an Elliptic Eigenvalue Problem with an Indefinite Weight on Domains in Noncompact Riemannian Manifolds

Abstract

The aim of this work is to establish sufficient conditions ensuring the continuity and compactness of the weighted Sobolev embedding and trace operators $$ W^{1, q}\left(\Omega ; \mathtt{V}_0, \mathtt{V}_1\right) \rightarrow L^{q_0}\left(\Omega ; \mathtt{V}_2\right) \quad \text { and } \quad W^{1, q}\left(\Omega ; \mathtt{V}_0, \mathtt{V}_1\right) \rightarrow L^{q_1}(\partial \Omega ; \mathtt{W}) . $$ As an application, we study the Neumann eigenvalue problem with an indefinite weight $$ \left\{\begin{aligned} -\operatorname{div}\left(\mathtt{V}_1 \nabla u\right)+\mathtt{V}_0 u & =\lambda \mathtt{V}_2 \tau u & & \text { in } \Omega, \\ \mathtt{V}_1 \frac{\partial u}{\partial \nu} & =0 & & \text { on } \partial \Omega . \end{aligned}\right. $$ where $\Omega$ is an open subset of a noncompact Riemannian manifold, $\lambda$ is a real number, $\tau$ is a sign-changing function, and $\mathtt{V}_0, \mathtt{V}_1, \mathtt{V}_2$, and $\mathtt{W}$ are weight functions satisfying suitable conditions. We prove that this problem has infinitely many positive and negative eigenvalues. We also establish boundedness of its weak eigenfunctions and, for $u \in W_0^{1,2}\left(\bar{\Omega} ; \mathtt{V}_0, \mathtt{V}_1\right)$, the decay property $$ \lim _{m \rightarrow \infty} \underset{\Omega \backslash \overline{D_m}}{\operatorname{ess} \sup }|u|=0 $$ where $\left(D_m\right)_{m \in \mathbb{N}}$ is an exhaustion of the manifold by bounded open sets.