On a class of critical double phase problems

Abstract

In this paper we study a class of double phase problems involving critical growth, namely $-\text{div}\big(|\nabla u|^{p-2} \nabla u+ \mu(x) |\nabla u|^{q-2} \nabla u\big)=\lambda|u|^{\vartheta-2}u+|u|^{p^*-2}u$ 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 $\lambda^*>0$ such that the problem above has infinitely many weak solutions with negative energy values for any $\lambda\in (0,\lambda^*)$.