Eigenvalues for anisotropic p-Laplacian under a Steklov-like boundary condition

Abstract

The eigenvalue problem $$-\mbox{div}~\Big(\frac{1}{p}\nabla_\xi \big(F^p\big (\nabla u)\Big)=\lambda a(x) \mid u\mid ^{q-2}u,$$ with $q\in (1, \infty),~ p\in \Big(\frac{Nq}{N+q-1}, \infty\Big),~ p\neq q,$ subject to Steklov-like boundary condition, $$F^{p-1}(\nabla u)\nabla _\xi F (\nabla u)\cdot \nu=\lambda b(x) \mid u\mid ^{q-2}u$$ is investigated on a bounded Lipschitz domain $\Omega\subset \mathbb{R}^ N,~N\geq 2$. Here, $F$ stands for a $C^2(\mathbb{R}^N\setminus \{0\})$ norm and $a\in L^{\infty}(\Omega),~ b\in L^{\infty}(\partial\Omega)$ are given nonnegative functions satisfying
\[
\int_\Omega a~dx+\int_{\partial\Omega} b~d\sigma >0.
\]
Using appropriate variational methods, we are able to prove that the set of eigenvalues of
this problem is the interval $[0, \infty)$.