Multi-valued variational inequalities in unbounded domains: existence, comparison and extremal solutions

Abstract

In this paper, we study multi-valued quasilinear elliptic variational inequalities of the form $u\in K:0\in -{\mathrm{\Delta }}_{p}u+aF(u)+\mathrm{\partial }{I}_K(u),$ in all of ${\mathbb{R}}^N$ as well as in the exterior domain $\mathrm{\Omega }={\mathbb{R}}^N\setminus \overline{B(0,1)}$, where ${\mathrm{\Delta }}_p$ is the p-Laplacian, K is a closed convex subset of the Beppo-Levi space $D^{1,p}({\mathbb{R}}^N)$ with 1<p<N, or $D^{1,N}(\mathrm{\Omega })$ with p = N, and $I_K$ is the indicator functional corresponding to K with its subdifferential $\mathrm{\partial }{I}_K$. The lower order multi-valued operator $F$ is generated by a multi-valued, upper semicontinuous function $f:\mathbb{R}\to 2^{\mathbb{R}}\setminus \{\mathrm{\varnothing }\}$, and the measurable coefficient a is supposed to decay like: $|a(x)|\le c\frac{1}{1+|x{|}^{N+\alpha }},\mathrm{\forall }x\in {\mathbb{R}}^N(\alpha >0).$ Our main goals are as follows. First, we provide a new approach to an existence theory for the above multi-valued variational inequalities in all ${\mathbb{R}}^N$ for 1<p<N as well as in the exterior domain Ω for the borderline case $p=N\ge 2$. Second, we establish an enclosure and comparison principle based on appropriately defined sub-supersolutions, and prove the existence of extremal solutions. Third, by means of the sub-supersolution method provided here, we show that certain rather general classes of variational-hemivariational type inequalities turn out to be only subclasses of the above class of multi-valued elliptic variational inequalities. Finally, we apply the abstract theory to a multi-valued obstacle problem.

Keywords:

• Variational inequality
• Beppo–Levi space
• upper semicontinuous multi-valued mapping
• extremal solution
• comparison
• variational-hemivariational inequalities

2020 Mathematics Subject Classifications:

• 35B51
• 35J87
• 49J52
• 49J53

Disclosure statement

No potential conflict of interest was reported by the author(s).