An Eigenvalue Problem for the Double Phase Differential Operator

We consider an eigenvalue problem for the nonautonomous double phase differential operator. We show that there exist two positive numbers \(0<\hat{\lambda }\le \lambda ^*\) $0<\widehat{\lambda }\le {\lambda }^{\ast }$ depending only on the data of the equation such that every \(\lambda \ge \lambda ^*\) $\lambda \ge {\lambda }^{\ast }$ is an eigenvalue, while \(0<\lambda <\hat{\lambda }\) $0<\lambda <\widehat{\lambda }$ is not an eigenvalue. It is an open problem if \(\hat{\lambda }=\lambda ^*\) $\widehat{\lambda }={\lambda }^{\ast }$ and also if \(\hat{\lambda }<\lambda ^*\) $\widehat{\lambda }<{\lambda }^{\ast }$ , then what can be said about the interval \([\hat{\lambda },\lambda ^*)\) $[\widehat{\lambda },{\lambda }^{\ast })$ , does it contain any eigenvalues?