On the eigenvalue set of the $(p,q)$-Laplacian with a Neumann-Steklov boundary condition

Abstract

Consider in a bounded domain $\Omega \subset \mathbb{R}^N$, $N\ge 2$, with smooth boundary $\partial \Omega$, the following eigenvalue problem\begin{eqnarray}& ~ & -\Delta_p u-\Delta_q u=\lambda a(x) | u | ^{r-2}u\ \ \mbox{ in}~ \Omega, \nonumber \\ & ~ & \big( | \nabla u | ^{p-2}+ | \nabla u | ^{q-2}\big)\frac{\partial u}{\partial\nu}=\lambda b(x) | u | ^{r-2}u~ \mbox{ on}~ \partial \Omega, \nonumber\end{eqnarray}where $1 < q < r < p < \infty,$ with $r < q(N-1)/(N-q)$ if $q < N$; $a\in L^{\infty}(\Omega),\ b\in L^{\infty}(\partial\Omega)$ are given nonnegativefunctions satisfying $\int_\Omega a~dx+\int_{\partial\Omega} b\,d\sigma > 0.$Under these assumptions, we prove that there exist two positive constants $\lambda_* < \lambda^*$ such that any $\lambda\in \{0\}\cup [\lambda^*, \infty)$ is an eigenvalue of this problem, while the set$(-\infty, 0)\cup (0, \lambda_*)$ contains no eigenvalue of the problem. This result is complementary to previous results related to the above eigenvalue problem.