Unbounded principal eigenfunctions for problems on all ${\bf R}^N$

Abstract

We investigate the existence of principal eigenvalues, i.e., values of $\lambda$ for which the corresponding eigenfunction is positive, for the problem $ - \Delta u(x) = \lambda g(x) u(x)$ for $x \in \mathbb R ^N$ where $g$ is a smooth function which may change sign. Unlike most previous studies the eigenfunction is not required to $\to 0$ as $|x| \to \infty$. It is shown that there may exist a closed interval of principal eigenvalues $[\lambda_* , \lambda^*]$ and sufficient conditions are given to ensure that principal eigenfunctions $\to 0$ as $|x| \to \infty$ if and only if $\lambda = \lambda^*$ or $\lambda_*$.