On the eigenvalue problem involving the weighted $p$-Laplacian in radially symmetric domains

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.