On a three critical points theorem

Abstract

In this paper, using a recent result by J. Saint Raymond ([6]), we improve the three critical points theorem established in [5]. Very recently,in [5], the following three critical points result has been established:Theorem A. Let X be a separable and reflexive real Banach space; F X X 3 R a continuously GaÃteaux differentiable and convex functional whose GaÃteaux derivative admits a continuous inverse on X à ; Y X X 3 R a continuously GaÃteaux differentiable functional whose GaÃteaux derivative is compact; I 7 R an interval; m 0 P I. Assume that lim kxk3 I Fx l À m 0 Yx I for all l P I, and that there exists a continuous concave function h X I 3 R such that sup lPI inf xPX Fx l À m 0 Yx hl`inf xPX sup lPI Fx l À m 0 Yx hl X Then, there exists l à P I nfm 0 g such that the equation F H x l à À m 0 Y H x 0 has at least three solutions in X.The aim of the present paper is to improve Theorem A in a twofold way. Namely, we wish to show that convexity of F can be replaced by sequential weak lower semicontinuity and, at the same time, that the conclusion even holds for each l à in an open sub-interval of I.So, we wish to prove the following Theorem 1. Let X be a separable and reflexive real Banach space; F X X 3 R a continuously GaÃteaux differentiable and sequentially weakly lower semicontinuous functional whose GaÃteaux derivative admits a continuous inverse on X à ; Y X X 3 R a continuously GaÃteaux differentiable functional whose GaÃteaux derivative is compact; I 7 R an interval. Assume that lim kxk3 I Fx lYx I Mathematics Subject Classification (1991): 35J20, 49J35, 58E05.

Higher nonlocal problems with bounded potential

“…We emphasize that in [11, Theorem 1.6] Ricceri established a theorem tailormade for a class of nonlocal problems involving nonlinearities with bounded primitive. This result follows from [10,Theorem 3] and the main novelty obtained in the most recent paper [11] is that, in contrast with a large part of the existing literature, the abstract energy functional does not depend on the parameter λ in an affine way.…”

Multiple solutions for a p-biharmonic equation with nonlinear boundary conditions

“…From (9), there is σ 0 > 0 such that u σ 0 2,p > 0 and Ψ(u σ 0 ) > 0. Now, by Lemma 3, we can choose ρ 0 ∈ (0, 1) such that…”

“…Indeed, if f ≡ 0 and g ≡ 0, then (H1) and (H2) clearly hold, but problem (1) has only the trivial solution. Theorem 1 will be proved by using a result on the existence of at least three critical points by Bonanno 8 which is a refinement of a general principle of Ricceri 9,10 . For the reader's convenience, we describe it as follows.…”

Multiple solutions for elliptic problems with singular and sublinear potentials

“…The purpose of this paper is to obtain multiple solutions for the problem Theorem 1 will be proved by means of a recent three critical point result of Bonanno [1] which is actually a refinement of a general principle of Ricceri [11,12]. Furthermore, in Section 3 we will give additional information as far as the localization of the interval Λ µ is concerned (see Remark 2).…”