On the eigenvalue set of the $(p,q)$-Laplacian with a Neumann-Steklov boundary condition
Luminiţa Barbu, Gheorghe Moroșanu · Differential and Integral Equations · 2023
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, onumber \\ & ~ & \big( | abla u | ^{p-2}+ | abla u | ^{q-2}\big)\frac{\partial u}{\partial u} =\lambda b(x) | u | ^ {r-2}u~ \mbox{ on}~ \partial \Omega, onumber \end{eqnarray} where $1 0.$ 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.