A further refinement of a three critical points theorem

In this paper, we complete the refinement process, made by Ricceri (2009) [4], of a result established by Ricceri (2000) [1], which is one of the most applied abstract multiplicity theorems in the past decade. A sample of application of our new result is as follows. Let Omega subset of R(n) (n >= 3) be a bounded domain with smooth boundary and let 1 < p < q < n+2/n-2. Then, for each is an element of > 0 small enough, there exists lambda(is an element of) > 0 such that, for every compact interval [a, b] subset of ]0, lambda(is an element of)[, there exists rho > 0 with the following property: for every lambda is an element of [a, b] and every continuous function h : R -> R satisfying lim sup(vertical bar xi vertical bar ->+infinity) vertical bar h(xi)vertical bar/vertical bar xi vertical bar(s) < + infinity for some s is an element of ]0, n+2/n-2 [, there exists delta > 0 such that, for each nu is an element of [0, delta], the problem {-Delta u = is an element of vertical bar u vertical bar(p-1)u - lambda vertical bar u vertical bar(q-1)u + nu h(u) in Omega u = 0 on partial derivative Omega has at least three weak solutions whose norms in H(0)(1)(Omega) are less than rho. (C) 2011 Elsevier Ltd. All rights reserved.