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On the eigenvalue problem involving the weighted \(p\)-Laplacian in radially symmetric domains. (English) Zbl 1401.35233
Summary: We investigate the following eigenvalue problem \[ \begin{cases} -\operatorname{div}(L(x) |\nabla u |^{p - 2}\nabla u) = \lambda K(x) | u |^{p - 2} u \quad &\text{in } A_{R_1}^{R_2}, \\ u = 0 & \text{on } \partial A_{R_1}^{R_2}, \end{cases} \] 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. The asymptotic estimates for \(u(x)\) and \(\nabla u(x)\) as \(| x | \rightarrow R_1^+\) or \(R_2^-\) are also investigated.
MSC:
35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs
35J62 Quasilinear elliptic equations
35B40 Asymptotic behavior of solutions to PDEs
35B50 Maximum principles in context of PDEs
35J20 Variational methods for second-order elliptic equations
35J25 Boundary value problems for second-order elliptic equations
35P15 Estimates of eigenvalues in context of PDEs
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