On a class of critical double phase problems

In this paper we study a class of double phase problems involving critical growth, namely $-\text{div}\big(|\nabla u|^{p-2} \nabla u+ μ(x) |\nabla u|^{q-2} \nabla u\big)=λ|u|^{\vartheta-2}u+|u|^{p^*-2}u$ in $Ω$ and $u= 0$ on $\partialΩ$, where $Ω\subset \mathbb{R}^N$ is a bounded Lipschitz domain, $1<\vartheta<p<q<N$, $\frac{q}{p}<1+\frac{1}{N}$ and $μ(\cdot)$ is a nonnegative Lipschitz continuous weight function. The operator involved is the so-called double phase operator, which reduces to the $p$-Laplacian or the $(p,q)$-Laplacian when $μ\equiv 0$ or $\inf μ>0$, respectively. Based on variational and topological tools such as truncation arguments and genus theory, we show the existence of $λ^*>0$ such that the problem above has infinitely many weak solutions with negative energy values for any $λ\in (0,λ^*)$.

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