Extremal solutions in systems of variational inequalities with multivalued mappings

ABSTRACT

In this paper, we use a sub-supersolution method to study systems of variational inequalities of the form: $\begin{array}{rl}& ⟨{\mathcal{A}}_k\left(u_k\right)+{\mathcal{F}}_k\left(u\right),v_k-u_k⟩\ge 0,\mathrm{\forall }{v}_k\in K_k\\ & u_k\in K_k,\end{array}$ $(k=1,\dots ,m)$, where ${\mathcal{A}}_k$ and ${\mathcal{F}}_k$ are multivalued mappings with possibly non-power growths and $K_k$ is a closed, convex set. We introduce a concept of mixed extremal solutions in the set-theoretic sense and prove the existence of such solutions between sub- and supersolutions. We also show the existence of least and greatest solutions of the above system between sub- and supersolutions if the lower order terms have certain increasing properties.

COMMUNICATED BY:

• C. Siegfried

Keywords:

• System of variational inequalities
• multivalued mapping
• sub-supersolution
• extremal solution
• Orlicz–Sobolev space

2000 Mathematics Subject Classifications:

• Primary: 35B45
• 35J65
• 35J60

Acknowledgements

The author thanks the referees for their valuable comments and suggestions.

Disclosure statement

No potential conflict of interest was reported by the author.