Abdelmajid Boukhsas, Brahim Ouhamou

Steklov Eigenvalues Problems for Generalized (p,q)-Laplacian Type Operators

abstract:
In this paper, we study the following class of $(p,q)$ elliptic problems under Steklov-type boundary conditions
\begin{equation*}
\begin{cases}
-\divv\big(a(|\nabla u|^{p})|\nabla u|^{p-2}\nabla u\big)+a(|u|^{p})|u|^{p-2}u=0 \;\;\text{in $\Omega$}, \\
a(|\nabla u|^{p})|\nabla u|^{p-2}\nabla u\cdot\nu=\lambda |u|^{m-2}u \;\;\text{on $\partial\Omega$},
\end{cases}
\end{equation*}
where $\Omega$ is a smooth bounded domain in $\mathbb{R}^N$ $( N \geq 2)$, $\nu$ is the outward unit normal vector on $\partial\Omega$, $2 \leq p < N$, $m\in\mathbb{R}$ with $m > 1$ in suitable ranges listed later and $a$ is a $C^1$ real function and $\lambda > 0$ is a real parameter. Using variational methods, we establish the existence of a continuous and unbounded set of positive generalized eigenvalues.

Mathematics Subject Classification: 35J20, 35J62, 35J70, 35P05, 35P30

Key words and phrases: $(p,q)$-Laplacian, nonlinear boundary conditions, Steklov eigenvalue problem, mountain pass theorem, Ekeland variational