We investigate the following eigenvalue problem \begin{ align*} \begin{ cases} -\operatorname{ div}\left( L(x) |\nabla u| ^{ p-2}\nabla u\right)=\lambda K(x)|u|^{ p-2}u \quad \text{ in } A_{ R_1}^{ R_2} , u=0\quad \text{ on } \partial A_{ R_1}^{ R_2} , \end{ cases} \end{ align*} where $A_{ R_1}^{ R_2}:=\{ x\in\mathbb{ R}^N: R_1<|x|<R_2\}$ $(0< R_1<R_2\leq\infty)$, $\lambda>0$ is a parameter, the weights $L$ and $K$ are measurable with $L$ positive a.e. in $A_{ R_1}^{ R_2}$ and $K$ possibly sign-changing in $A_{ R_1}^{ R_2}$. We prove the existence of the first eigenpair and discuss the regularity and positiveness of eigenfunctions. %apriori bounds of any eigenfunction as well as local boundedness. The asymptotic estimates for $u(x)$ and $\nabla u(x)$ as $|x|\to R_1^+$ or $R_2^-$ are also investigated.
Categories: math.AP