Infinitely Many Solutions for a Boundary Value Problem with Discontinuous Nonlinearities

Abstract

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.

Positive Solutions of Singular Dirichlet Problems via Variational Methods

“…Theorem 2.4 ( [8,9]). Let X be a reflexive real Banach space; let Φ, Ψ : X → R be two Gâteaux differentiable functionals such that Φ is sequentially weakly lower semicontinuous, strongly continuous, and coercive and Ψ is sequentially weakly upper semicontinuous.…”

Infinitely many weak solutions for fourth-order equations depending on two parameters

“…Recently in [6], presenting a version of the infinitely many critical points theorem of Ricceri (see [17,Theorem 2.5]), the existence of an unbounded sequence of weak solutions for a Strum-Liouville problem, having discontinuous nonlinearities, has been established. In a such approach, an appropriate oscillating behavior of the nonlinear term either at infinity or at zero is required.…”

Infinitely Many Solutions for a Perturbed Quasilinear Two-Point Boundary Value Problem

“…Consider the following perturbed quasilinear two-point boundary value problem on a bounded interval [a, b] in R (a < b) Employing a smooth version of Theorem 2.1 of [7] which is a more precise version of Ricceri's Variational Principle [30,Theorem 2.5] (see Theorem 1.2), under some appropriate hypotheses on the behavior of the potential of f , under some conditions on the potentials of p, g and h, at infinity, we establish the existence of a precise interval of parameters Λ such that, for each λ ∈ Λ, the problem (1) admits a sequence of weak solutions which are unbounded in the Sobolev space W 1,2 0 ([a, b]); this is done in Theorem 2.1. We also list some special cases of Theorem 2.1.…”

“…For a discussion about the existence of infinitely many solutions for differential equations, using Ricceri's Variational Principle [30] and its variants ( [7] and [27]), we refer the reader to the papers [3,4,8,9,10,11,13,14,15,18,19,21,25,31]. We also refer the reader to the papers [26,28,29] where the existence of infinitely many solutions for some boundary value problems has been studied by using different approaches.…”

“…Our main tool is the celebrated Ricceri's Variational Principle [30, Theorem 2.5] that we now recall as given by Bonanno and Molica Bisci in [7]: Theorem 1.2. Let X be a reflexive real Banach space, let Φ, Ψ : X −→ R be two Gâteaux differentiable functionals such that Φ is sequentially weakly lower semicontinuous, strongly continuous, and coercive and Ψ is sequentially weakly upper semicontinuous.…”