Existence results for Dirichlet double phase differential inclusions

Abstract

In this paper we consider a class of double phase differential inclusions of the type
$$
\left\{
\begin{array}{ll}
-{\rm div\;}\left(|\nabla u|^{p-2}\nabla u+\mu(x)|\nabla u|^{q-2}\nabla u\right) \in \partial_C^2 f(x,u) , & \mbox{ in }\Omega,\\
u=0, & \mbox{ on }\partial\Omega,
\end{array}
\right.
$$
where \(\Omega \subset \mathbb{R}^N\) with \(N\ge 2\) is a bounded domain with Lipschitz boundary, \(f(x,t)\) is measurable w.r.t. the first variable on \(\Omega\) and locally Lipschitz w.r.t. the second variable and \(\partial_C^2 f(x,\cdot)\) stands for the Clarke subdifferential of \(t\mapsto f(x,t)\). The variational formulation of the problem gives rise to a so-called hemivariational inequality and the corresponding energy functional is not differentiable, but only locally Lipschitz. We use nonsmooth critical point theory to prove the existence of at least one weak solution, provided the \(\partial_C^2 f(x,\cdot)\) satisfies an appropriate growth condition.