Infinitely Many Solutions for a Boundary Value Problem with Discontinuous Nonlinearities

Hindawi Publishing Corporation Boundary Value Problems Volume 2009, Article ID 670675, 20 pages doi:10.1155/2009/670675 Research Article Infinitely Many Solutions for a Boundary Value Problem with Discontinuous Nonlinearities Gabriele Bonanno1 and Giovanni Molica Bisci2 1 Mathematics Section, Department of Science for Engineering and Architecture, Engineering Faculty, University of Messina, 98166 Messina, Italy 2 PAU Department, Architecture Faculty, University of Reggio, Calabria, 89100 Reggio Calabria, Italy Correspondence should be addressed to Gabriele Bonanno, Received 16 October 2008; Accepted 11 February 2009 Recommended by Ivan T. Kiguradze 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. Copyright q 2009 G. Bonanno and G. M. Bisci. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction The aim of this paper is to establish infinitely many solutions for two-point boundary value problems with the nonlinear term possibly discontinuous. We immediately emphasize the following theorem which is a particular case of our main result Theorem 3.1. Theorem 1.1. Let f : R → R be a locally bounded, and almost everywhere continuous function such ξ that infR f > 0. Put Fξ : 0 ftdt for every ξ ∈ R and assume that Fξ 1 Fξ < lim sup 2 . 2 ξ → ∞ 4 ξ → ∞ ξ ξ lim inf 1.1 Then, for each λ ∈8/lim supξ → ∞ Fξ/ξ2 , 2/lim infξ → ∞ Fξ/ξ2 , the problem −u  λfu in 0, 1 u0  u1  0 admits a sequence of pairwise distinct positive weak solutions. G1,0  f,λ 2 Boundary Value Problems Clearly, when f is continuous in R, the solutions in Theorem 1.1 are classical in this case, it is enough to assume infR f ≥ 0; see Corollary 3.5. Moreover, substituting ξ → ∞ with ξ → 0 , the same results hold and, in addition, the sequence of pairwise distinct positive solutions uniformly converges to zero see Theorem 3.9 and Corollary 3.10. When f is a continuous function, results of the existence of infinitely many solutions  are obtained, for example, in 1–7. We observe that in the very interesting for problem G1,0 f,λ paper 6, the authors assume lim infξ → ∞ Fξ/ξ2  0 and lim supξ → ∞ Fξ/ξ2  ∞, which are conditions that imply our key assumption. Very recently, in 4, a more general condition than the previous assumption has been assumed, requiring in addition, however, that limξ → ∞ fξ  ∞. Moreover, we also observe that the results in 1, 2 are obtained by using the important Variational Principle of Ricceri 8, which is, basically, the same as our tool. We emphasize that, also when f is a continuos function, our theorems in this paper and the results in 1–7 are mutually independent see Remark 3.13 and Examples 3.11 and 3.12. When the nonlinear term f is discontinuous, there have been many approaches to studying a nonlinear eigenvalue differential equation as it arises in physics problems, such as nonlinear elasticity theory, and mechanics, and engineering topics. Chang in 9 established the critical point theory for nondifferentiable functionals and presented some applications to partial differential equations with discontinuous nonlinearities. Next, Motreanu and Panagiotopoulos see 10, Chapter 3 studied the critical point theory for non-smooth functionals and in this framework, very recently, Marano and Motreanu, in 11, obtained an infinitely many critical points theorem, which extends the Variational Principle of Ricceri to non-smooth functionals, and applies this result to variational-hemivariational inequalities and semilinear elliptic eigenvalue problems with discontinuous nonlinearities. In this paper, we present a more precise version of the infinitely many critical points theorem of Marano and Motreanu Theorem 2.1, obtained by a completely different proof see Remark 2.2 and, by using the previous theorem, we establish our main result Theorem 3.1 on the existence of infinitely many solutions for a two-point boundary value problem with the Sturm-Liouville equation having discontinuous nonlinear term. We explicitly observe that methods and techniques used in the proof of Theorem 3.1 can be applied to a wide class of nonlinear differential problems to investigate infinitely many solutions. The note is arranged as follows. In Section 2, we recall some basic definitions and our abstract framework, while Section 3 is devoted to infinitely many solutions for the SturmLiouville problem. Finally, we point out that the existence of multiple solutions for nonlinear differential problems has been studied in several papers by using different techniques see, e.g., 12, 13 and references therein. 2. Infinitely Many Critical Points Let X,  ·  be a real Banach space. We denote by X ∗ the dual space of X, while ·, · stands for the duality pairing between X ∗ and X. A function Φ : X → R is called locally Lipschitz continuous when, to every x ∈ X, there corresponds a neighbourhood Vx of x and a constant Lx ≥ 0 such that   Φz − Φw ≤ Lx z − w ∀z, w ∈ Vx . 2.1 Boundary Value Problems 3 If x, z ∈ X, we write Φ◦ x; z for the generalized directional derivative of Φ at the point x along the direction z, that is, Φ◦ x; z : lim sup w → x, t → 0 Φw  tz − Φw . t 2.2 The generalized gradient of the function Φ in x, denoted by ∂Φx, is the set     ∂Φx : x∗ ∈ X ∗ : x∗ , z ≤ Φ◦ x; z ∀z ∈ X . 2.3 We say that x ∈ X is a generalized critical point of Φ when Φ◦ x; z ≥ 0 ∀z ∈ X, 2.4 that clearly signifies 0 ∈ ∂Φx. When a non-smooth functional, Ψ : X →  − ∞, ∞, is expressed as a sum of a locally Lipschitz function, Φ : X → R, and a convex, proper, and lower semicontinuous function, j : X →  − ∞, ∞, that is Ψ : Φ  j, a generalized critical point of Ψ is every u ∈ X such that Φ◦ u; v − u  jv − ju ≥ 0 2.5 for all v ∈ X see 10, Chapter 3. Here, and in the sequel, X is a reflexive real Banach space, Φ : X → R is a sequentially weakly lower semicontinuous functional, Υ : X → R is a sequentially weakly upper semicontinuous functional, λ is a positive real parameter, j : X →  − ∞, ∞ is a convex, proper and lower semicontinuous functional and Dj is the effective dominion of j. Write Ψ : Υ − j, Iλ : Φ − λΨ  Φ − λΥ  λj. 2.6 We also assume that Φ is coercive and  Dj ∩ Φ−1  − ∞, r / ∅ 2.7 for all r > infX Φ. Moreover, owing to 2.7 and provided r > infX Φ, we can define  ϕr  inf supu∈Φ−1 −∞,r Ψu − Ψu u∈Φ−1 −∞,r γ : lim (...truncated)