Continuous spectrum for a degenerate eigenvalue problem with (p2, p2)-growth

Abstract

In this paper we consider an eigenvalue problem driven by two non-homogeneous differential operators with variable $(p_2, p_2)$-growth. We establish that for $\lambda_1 > 0$, any $\lambda \in [\lambda_1, \infty)$ is an eigenvalue; moreover, for a positive constant $\lambda_0 \leq \lambda_1$, we find the nonexistence of eigenvalues in $(0, \lambda_0)$. The proof is based on variational arguments and a Caffarelli--Kohn--Nirenberg-type inequality with variable exponent.