Asymptotical Multiplicity and Some Reversed Variational Inequalities

We are concerned with multiplicity results for solutions of some reversed variational inequalities, in which the inequality is opposite with respect to the classical inequalities introduced by Lions and Stampacchia. The inequalities we study arise from a family (Pω) of elliptic problems of the fourth order when ω tends to ∞. We use two basic tools: the ∇theorems and a theorem about the multiplicity of “asymptotically critical” points. In the last section some open problems are listed. 1. Introducing the problem Some sequences of elliptic problems lead in a natural way to study some variational inequalities whose sign is opposite to the one of the usual inequalities of Lions–Stampacchia’s type (see [8]). For this reason we call them “reversed” variational inequalities and, at least for the moment, we have found several difficulties in finding their deep sense. In short, we have many questions and few answers. In order to present a possible genesis of the reversed variational inequalities, let us consider, for example, the bounce problem: if an open subset Ω of R represents the “billiard” and V is the potential energy of a conservative force field in R , we can think to obtain the bounce trajectories γ: [0, 1]→ Ω between 2000 Mathematics Subject Classification. Primary 49J40; Secondary 58E05, 58E35.