Variational approach to noncoercive systems of elliptic variational inequalities via trapping region†

Abstract

This article deals with a system of quasilinear elliptic variational inequalities whose leading differential operator is a diagonal (p ₁, p ₂)-Laplace operator with 1 < p ₁, p ₂ < ∞, and whose lower order vector field f = (f ₁, f ₂) is a gradient field, which is not subject to any growth restriction, in particular, it may have supercritical growth, and thus coercivity is violated. The novelty of this article is to establish a variational approach of the, in general, noncoercive elliptic system of variational inequalities by introducing the concept of trapping region for such type of problems. The trapping region allows us to transform the given noncoercive system of variational inequalities into an associated ‘truncated’ system to which variational methods can be applied. We are going to prove that the ‘truncated’ system possesses solutions, which can be characterized as the critical points of a suitably constructed (nonsmooth) energy functional, and any critical point is shown to be a solution of the original problem within the trapping region. Moreover, applications to quasilinear elliptic systems under Dirichlet boundary conditions as well as to elliptic systems under obstacle constraints are treated by the theory developed in this article.

†Dedicated to Professor R.P. Gilbert on the occasion of his 80th birthday.

Keywords:

• elliptic system of variational inequalities
• trapping region
• pairs of coupled sub-supersolutions
• system of obstacle problems
• variational method
• critical point
• comparison principle

AMS Subject Classifications::

• 35B38
• 35J88
• 49J40

Acknowledgements

The author would like to thank the referees for their careful reading and helpful comments.

Notes

†Dedicated to Professor R.P. Gilbert on the occasion of his 80th birthday.