Nonlinear double phase eigenvalue problems

Abstract

We consider a nonlinear eigenvalue problem driven by the double phase differential operator. We prove two existence theorems, both producing a continuous spectrum and the first generates eigenfunctions which blow up in the $W_0^{1,\theta }(\mathrm{\Omega })\cap L^r(\mathrm{\Omega })$ -norm $(1<q<p<r<q^{\ast })$ , while the second generates eigenfunctions which vanish in the $W_0^{1,\theta }(\mathrm{\Omega })\cap L^{\infty }(\mathrm{\Omega })$ -norm as $\lambda \to 0^{+}$ .