Gabriele Bonanno

Some remarks on a three critical points theorem

Abstract Some remarks on a strict minimax inequality, which plays a fundamental role in Ricceris three critical points theorem, are presented. As a consequence, some recent applications of Ricceris theorem to nonlinear boundary value problems are revisited by obtaining more precise conclusions.

Infinitely Many Solutions for a Boundary Value Problem with Discontinuous Nonlinearities

The existence of infinitely many solutions for a Sturm-Liouville boundary value problem, under an appropriate oscillating behavior of the possibly discontinuous nonlinear term, is obtained. Several special cases and consequences are pointed out and some examples are presented. The technical approach is mainly based on a result of infinitely many critical points for locally Lipschitz functions.

On the structure of the critical set of non-differentiable functions with a weak compactness condition

The nature of critical points generated by a Ghoussoubs type min–max principle for locally Lipschitz continuous functionals fulfilling a weak Palais–Smale assumption, which contains the so-called non-smooth Cerami condition, is investigated. Two meaningful special cases are then pointed out; see Theorems 3.5 and 3.6.

A three critical points theorem and its applications to the ordinary Dirichlet problem

The aim of this paper is twofold. On one hand we establish a three critical points theorem for functionals depending on a real parameter

Multiplicity theorems for the Dirichlet problem involving the p-Laplacian

\lambda \in \Lambda

A minimax inequality and its applications to ordinary differential equations

, which is different from the one proved by B. Ricceri in [Arch. Math. {\bf 75} (2000), 220-226] and gives an estimate of where

Existence of solutions to boundary value problem for impulsive fractional differential equations

\Lambda

Infinitely many solutions for a Dirichlet problem involving the p-Laplacian

can be located. On the other hand, as an application of the previous result, we prove an existence theorem of three classical solutions for a two-point boundary value problem which is independent from the one by J. Henderson and H. B. Thompson [J. Differential Equations {\bf 166} (2000), 443-454]. Specifically, an example is given where the key assumption of [J. Differential Equations {\bf 166} (2000), 443-454] fails. Nevertheless, the existence of three solutions can still be deduced using our theorem.

A Critical Points Theorem and Nonlinear Differential Problems

Abstract In this paper, we establish some results on the existence of at least three weak solutions for Dirichlet problems involving the p -Laplacian by a variational approach.

Infinitely many solutions for a class of discrete non-linear boundary value problems

The aim of this paper is to investigate the minimax inequality which plays a fundamental role in the critical points theorem of B. Ricceri below. Equivalent formulations are shown, and characterization is proved in particular for a special class of functionals. As an application, a multiplicity result for an ordinary Dirichlet problem is emphasized.