In this paper, we consider eigenvalues to the following double phase problem with unbalanced growth and indefinite weight, $$ -\Delta_p^a u-\Delta_q u =\lambda m(x) |u|^{ q-2}u \quad \mbox{ in} \,\, \R^N, $$ where { $N \geq 2$}, { $1<p, q<N$, $p \neq q$}, ${ a \in C^{ 0, 1}(\R^N, [0, +\infty))}$, $a \not\equiv 0$ and $m: \R^N \to \R$ 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 $\Delta_p^a u:=\textnormal{ div}(a(x) |\nabla u|^{ p-2} \nabla u)$. The problem can be degenerate, in the sense that the infimum of $a$ in $\R^N$ 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.
Categories: math.AP