On a class of critical double phase problems

Abstract

In this paper we study a class of double phase problems involving critical growth, namely $-\text{div}(|\nabla u{|}^{p-2}\nabla u+\mu (x)|\nabla u{|}^{q-2}\nabla u)=\lambda |u{|}^{\vartheta -2}u+|u{|}^{p^{\ast }-2}u$ in $\Omega$ and $u=0$ on $\partial \Omega$, where $\Omega \subset {\mathbb{R}}^N$ is a bounded Lipschitz domain, $1<\vartheta <p<q<N$, $\frac{q}{p}<1+\frac{1}{N}$ and $\mu (\cdot )$ 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 $\mu \equiv 0$ or $\inf \mu >0$, respectively. Based on variational and topological tools such as truncation arguments and genus theory, we show the existence of ${\lambda }^{\ast }>0$ such that the problem above has infinitely many weak solutions with negative energy values for any $\lambda \in (0,{\lambda }^{\ast })$.