| Summary: | We study the following class of double-phase nonlinear eigenvalue problems
$$
-\operatorname{div}\left[\phi(x,|\nabla u|)\nabla u+\psi(x,|\nabla u|)\nabla u\right]=\lambda f(x,u)
$$
in $\Omega$, $u=0$ on $\partial\Omega$, where $\Omega$ is a bounded domain from $\mathbb{R}^N$ and the potential functions $\phi$ and $\psi$ have $(p_1(x);p_2(x))$ variable growth. The primitive of the reaction term of the problem (the right-hand side) has indefinite sign in the variable $u$ and allows us to study functions with slower growth near $+\infty$, that is, it does not satisfy the Ambrosetti–Rabinowitz condition. Under these hypotheses we prove that for every parameter $\lambda\in \mathbb{R}^*_+$, the problem has an unbounded sequence of weak solutions. The proofs rely on variational arguments based on energy estimates and the use of Fountain Theorem.
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