A variational principle for problems with a hint of convexity

Abstract

A variational principle is introduced to provide a new formulation and resolution for several boundary value problems with a variational structure. This principle allows one to deal with problems well beyond the weakly compact structure. As a result, we study several super-critical semilinear Elliptic problems.

Multiplicity results for elliptic problems with super-critical concave and convex nonlinearties

“…It follows from (ii) in the theorem that −∆v = DΦ(ū). Thus, it follows from inequality (13) with v =v that…”

“…We shall be proving Theorems 1.1 and 1.2 by making use of a new abstract variational principle established recently in [13,14] (see also [11,12] for some new variational principles and [5] for an application in supercritical Neumann problems). To be more specific, let V be a reflexive Banach space, V * its topological dual and let K be a convex and weakly closed subset of V .…”

“…We shall now recall the following variational principle established recently in [13]. (ii) there exists v 0 ∈ K such that DΨ(v 0 ) = DΦ(u 0 ).…”

Existence results for a supercritical Neumann problem with a convex–concave non-linearity

“…We shall now recall the following variational principle recently established in [26] (see also [27]).…”

“…To prove Theorem 1.1, we utilize an abstract variational principle from [26] (see also [27]). To be more specific, let V be a reflexive Banach space, V * its topological dual and let K be a non-empty convex and weakly closed subset of V .…”