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

In this paper, we consider eigenvalues to the following double phase problem with unbalanced growth and indefinite weight,-Delta pau-Delta qu=lambda m(x)|u|q-2uinRN,$$\begin{equation*} \hspace*{3pc}-\Delta _pa u-\Delta _q u =\lambda m(x)|u|{q-2}u \quad \mbox{in} \,\, \mathbb {R}<^>N, \end{equation*}$$where N > 2$N \geqslant 2$, 1{0, 1}(\mathbb {R}N, [0, +\infty))$, a not equivalent to 0$a \not\equiv 0$ and m:RN -> R$m: \mathbb {R}N \rightarrow \mathbb {R}$ is an indefinite sign weight which may admit non-trivial positive and negative parts. Here, Delta q$\Delta _q$ is the q$q$-Laplacian operator and Delta pa$\Delta _pa$ is the weighted p$p$-Laplace operator defined by Delta pau:=div(a(x)| backward difference u|p-2 backward difference u)$\Delta _pa u:=\textnormal {div}(a(x)|\nabla u|{p-2} \nabla u)$. The problem can be degenerate, in the sense that the infimum of a$a$ in RN$\mathbb {R}N$ may be zero. Our main results distinguish between the cases p