positive eigenfunctions of a schrödinger operator

Abstract

the paper considers the eigenvalue problem \[ -\delta u-\alpha u+\lambda g(x)u=0\quad \mbox{with}u\in h^{1}(\mathbb{r}^{n}),\;u\neq0, \] where $\alpha,\lambda\in\mathbb{r}$ and \[g(x)\equiv0\mbox{on}\overline{\omega},\quad\ g(x)\in(0,1]\mbox{on}{\mathbb{r}^{n}}\setminus{\overline{\omega}}\quad\mbox{and}\quad\lim_{|x|\rightarrow+\infty}g(x)=1 \] for some bounded open set $\omega\in\mathbb{r}^{n}$.

given $\alpha>0$, does there exist a value of $\lambda>0$ for which the problem has a positive solution? it is shown that this occurs if and only if $\alpha$ lies in a certain interval $(\gamma,\xi_{1})$ and that in this case the value of $\lambda$ is unique, $\lambda=\lambda(\alpha)$. the properties of the function $\lambda(\alpha)$ are also discussed.