Continuous spectrum for a double-phase unbalanced growth eigenvalue problem (Q6566701)

Let \(\Omega \subseteq \mathbb{R}^N\) be a bounded domain with a Lipschitz boundary \(\partial \Omega\). The authors consider the following parametric double phase Dirichlet problem \[ -\Delta_{p}^{a} u-\Delta_{q} u=\lambda a(z)|u|^{p-2}u \quad\text{in } \Omega, \\\Nu \Big|_{\partial \Omega} =0, \, 1<q<p<N, \, \lambda>0, \] where \(a \in L^\infty(\Omega)\) with \(a(z)\geq 0\) for a.a. \(z \in \Omega\), and \(\Delta_{p}^a\) is the weighted \(p\)-Laplace differential operator given as \[\Delta_{p}^a u= \text{div }(a(z)|\nabla u|^{p-2}\nabla u).\]\N\NThe authors prove that the above problem has a continuous spectrum. In details, for all \(\lambda > \overline{\lambda}\), the problem admits a nontrivial solution, hence for \(0 \leq \lambda \leq \overline{\lambda}\) there is no nontrivial solution. Further, the critical value \(\overline{\lambda}\) is given in terms of the spectrum of \(\Delta_{p}^a\). The authors study the energy functional associated with the Dirichlet problem, and the approach is based on the Nehari method.