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

We prove existence and comparison results for multi-valued variational inequalities in a bounded domain $$\Omega $$ Ω of the form $$\begin{aligned} u\in K{:}\, 0 \in Au+\partial I_K(u)+{\mathcal {F}}(u)+{\mathcal {F}}_\Gamma (u)\quad \text {in }W^{1, {\mathcal {H}}}(\Omega )^*, \end{aligned}$$ u ∈ K : 0 ∈ A u + ∂ I K ( u ) + F ( u ) + F Γ ( u ) in W 1 , H ( Ω ) ∗ , where $$A{:}\,W^{1, {\mathcal {H}}}(\Omega ) \rightarrow W^{1, {\mathcal {H}}}(\Omega )^*$$ A : W 1 , H ( Ω ) → W 1 , H ( Ω ) ∗ given by $$\begin{aligned} Au:=-\text {div}\left( |\nabla u|^{p(x)-2} \nabla u+ \mu (x) |\nabla u|^{q(x)-2} \nabla u\right) \end{aligned}$$ A u : = - div | ∇ u | p ( x ) - 2 ∇ u + μ ( x ) | ∇ u | q ( x ) - 2 ∇ u for $$u \in W^{1, {\mathcal {H}}}(\Omega )$$ u ∈ W 1 , H ( Ω ) , is the double phase operator with variable exponents and $$W^{1, {\mathcal {H}}}(\Omega )$$ W 1 , H ( Ω ) 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–super-solution for the multi-valued variational inequality in the space $$W^{1, {\mathcal {H}}}(\Omega )$$ W 1 , H ( Ω ) based on appropriately defined sub- and super-solutions, which yields the existence of solutions within an ordered interval of sub–super-solution. Moreover, the existence of extremal solutions will be shown provided the closed, convex subset K of $$W^{1, {\mathcal {H}}}(\Omega )$$ W 1 , H ( Ω ) satisfies a lattice condition. As an application of the sub–super-solution 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 $${\mathcal {F}}, {\mathcal {F}}_\Gamma $$ F , F Γ need not be continuous. At the end, we give an example of the construction of sub- and supersolutions related to the problem above.