An eigenvalue problem for a variable exponent problem, via topological degree

The problem$ \begin{equation*} \left\{ \begin{array}{l} -\Delta_{p(|x|)} u - \Delta_{q} u = \lambda (|u|^{p(|x|)-2}u + |u|^{q-2}u) \quad \mbox{in} \ \mathcal B , \\ u = 0 \quad \mbox{on} \ \partial \mathcal B \end{array} \right. \end{equation*} $is considered, where $ \mathcal B = \{ x \in \mathbb{R}^N : |x|1 $ on $ [0, R] $, $ \Delta_{q} u = \mbox{div}(|\nabla u|^{q-2}\nabla u) $, and $ q>1 $. The existence of positive solutions is proved for every $ \lambda>\lambda_1(q) $, where $ \lambda_1(q) $ is the first eigenvalue of $ q $-Laplacian.