Multi-valued variational inequalities for variable exponent double phase problems: comparison and extremality results

TL;DR

This paper establishes existence, comparison, and extremality results for multi-valued variational inequalities involving a variable exponent double phase operator, even when coercivity conditions are not satisfied, using sub-supersolution methods.

Contribution

It introduces a sub-supersolution framework for multi-valued variational inequalities with variable exponent double phase operators, including extremal solutions and applications to hemivariational inequalities.

Findings

Existence of solutions under weak coercivity conditions.

Development of a sub-supersolution method for non-coercive problems.

Inclusion of generalized variational-hemivariational inequalities as special cases.

Abstract

We prove existence and comparison results for multi-valued variational inequalities in a bounded domain $\Omega$ of the form \begin{equation*} u\in K\,:\, 0 \in Au+\partial I_K(u)+\mathcal{F}(u)+\mathcal{F}_\Gamma(u)\quad\text{in }W^{1,\mathcal{H}}(\Omega)^*, \end{equation*} where $A\colon W^{1, \mathcal{H}}(\Omega) \to W^{1, \mathcal{H}}(\Omega)^*$ given by \begin{equation*} Au:=-\text{div}\left(|\nabla u|^{p(x)-2} \nabla u+ \mu(x) |\nabla u|^{q(x)-2} \nabla u\right) \end{equation*} for $u \in W^{1, \mathcal{H}}(\Omega)$, is the double phase operator with variable exponents and $W^{1, \mathcal{H}}(\Omega)$ is the associated Musielak-Orlicz Sobolev space. First, an existence result is proved under some weak coercivity condition. Our main focus aims at the treatment of the problem under consideration when coercivity fails. To this end we establish the method of sub-supersolution for the multi-valued variational inequality in the space $W^{1, \mathcal{H}}(\Omega)$ based on appropriately defined sub- and supersolutions, which yields the existence of solutions within an ordered interval of sub-supersolution. Moreover, the existence of extremal solutions will be shown provided the closed, convex subset $K$ of $W^{1, \mathcal{H}}(\Omega)$ satisfies a lattice condition. As an application of the sub-supersolution method we are able to show that a class of generalized variational-hemivariational inequalities with a leading double phase operator are included as a special case of the multi-valued variational inequality considered here. Based on a fixed point argument, we also study the case when the corresponding Nemytskij operators $\mathcal{F}, \mathcal{F}_\Gamma$ need not be continuous. At the end, we give a nontrivial example of the construction of sub- and supersolutions related to the problem above.