On a class of critical double phase problems
Abstract
In this paper we study a class of double phase problems involving critical growth, namely $\text{div}\big(\nabla u^{p2} \nabla u+ \mu(x) \nabla u^{q2} \nabla u\big)=\lambdau^{\vartheta2}u+u^{p^*2}u$ 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 socalled 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 $\lambda\in (0,\lambda^*)$.
 Publication:

arXiv eprints
 Pub Date:
 July 2021
 arXiv:
 arXiv:2107.12835
 Bibcode:
 2021arXiv210712835F
 Keywords:

 Mathematics  Analysis of PDEs