Non-autonomous double phase eigenvalue problems with indefinite weight and lack of compactness

Abstract

In this paper, we consider eigenvalues to the following double phase problem with unbalanced growth and indefinite weight, $$ -Δ_p^a u-Δ_q u =λm(x) |u|^{q-2}u \quad \mbox{in} \,\, \R^N, $$ where {$N\ge 2$}, {$1<p,q<N$, $p\ne q$}, $a\in C^{0,1}({\mathbb{R}}^N,[0,+\infty ))$, $a\not\equiv 0$ and $m:{\mathbb{R}}^N\to \mathbb{R}$ is {an indefinite sign weight which may admit nontrivial positive and negative parts}. Here ${\Delta }_q$ is the $q$-Laplacian operator and ${\Delta }_p^a$ is the weighted $p$-Laplace operator defined by ${\Delta }_p^{a}u:=\text{div}(a(x)|\nabla u{|}^{p-2}\nabla u)$. The problem can be degenerate, in the sense that the infimum of $a$ in ${\mathbb{R}}^N$ may be zero. Our main results distinguish between the cases $p<q$ and $q<p$. In the first case, we establish the existence of a {\it continuous} family of eigenvalues, starting from the principal frequency of a suitable single phase eigenvalue problem. In the latter case, we prove the existence of a {\it discrete} family of positive eigenvalues, which diverges to infinity.

16 pages