Critical Points and Applications to Nonlinear Problems

Gabriele Bonanno

Dipartimento di Informatica, Matematica, Elettronica e Trasporti Facoltà di Ingegneria - Università di Reggio Calabria via Graziella (Feo di Vito), 89100 Reggio Calabria, Italy bonanno@ing.unirc.it Key words and phrases: Critical points, variational methods, minimax inequality, three solutions, two point boundary value problem. 2000 Mathematics Subject Classification: 58E05; 35B38; 49J35; 34B15.

ABSTRACT Let X be a real Banach space and Φ, Ψ : X → IR two Gâteaux differentiable functions. Recently in [15], B.Ricceri

established, under suitable assumptions on X, Φ and Ψ, the existence of a real interval Λ such that, for each λ ∈ Λ, the functional

Φ + λΨ  (1)

admits three critical points in X whose norms are uniformly bounded in respect to λ. Several applications of Ricceri’s three critical points theorem to nonlinear boundary value problems have been given in [2]-[6], [8], [11], [15], [16] (see also [12] for the non-smooth case) establishing results of multiple solutions for equations depending on a parameter λ.

One of our aims is to present three critical points theorems, Theorem 2.1 and Theorem 2.2 of [7], which are consequences of Ricceri’s Theorem and, in addition, ensure an upper bound for Λ, without losing the uniform boundedness of norms of critical points of (1) in respect to λ. The proof of these theorems is based on some remarks on a suitable minimax inequality (see Section 2 of [3] and Section 1 of [7]) which is the key assumption of Ricceri’s Theorem.

Our other aim is to present a further three critical points theorem, Theorem 2.1 of [1], which is obtained with a completely different technique from the one mentioned above and ensures a lower bound for the interval Λ. Its proof is based on the variational principle of B.Ricceri ([14]) and on the mountain pass theorem as given by P.Pucci and J.Serrin in [13].

As an example of applications of the previous results to nonlinear differential problems we consider the following two point boundary value problem

\{ \begin{matrix}−u′′ = λf(u) \\ u(0) = u(1) = & 0 \end{matrix}  (2)

and we present Theorem 3.1 of [7] (see also Theorem 2 of [2]) and Theorem 3.1 of [1]. Here, as a way of example, we point out a particular case of both previous results.

Theorem 1 - Let f : IR → IR be a continuous function. Put F (t) = ∫_{0}^{t} f(ξ)dξ for every t ∈ IR and assume that there exist four positive constants c, d, b, s, with c < d and s < 2, such that

(i) ∫_{0}^{d} F (t)dt ≥ 0

(ii)  F (t) < \frac{1}{6}(\frac{c}{d})^{2}F (d)  for every t ∈ [−c, c]
(iii)  F (t) ≤ b(1 + |t|^{s})  for every t ∈ IR.

Then, for each λ ∈ ]\frac{8d^{2}}{F (d) + \frac{1}{d} ∫_{0}^{d} F (t)dt − 2max_{|t|≤c} F (t)}, \frac{2c^{2}}{max_{|t|≤c} F (t)}[, the problem (2) admits at least three solutions

in C^{2}([0, 1]) and, moreover, there exists an open interval Λ ⊆ ]0, _{\frac{4}{\frac{1}{4} \frac{F (d)}{d^{2}} − \frac{max_{|t|≤c} F (t)}{c^{2}}}} [ and a real positive number σ

such that, for each λ ∈ Λ, the problem (2) admits at least three solutions in C^{2}([0, 1]) whose norms in C^{2}([0, 1]) are less than σ .

There is a very wide literature on multiple solutions for two point boundary value problems and we refer to [9],

[10] and references therein for more details.

REFERENCES

[1] D.Averna and G.Bonanno, A three critical points theorem and its applications to the ordinary Dirichlet problem, preprint.

[2] G.Bonanno, Existence of three solutions for a two point boundary value problem, Appl. Math. Lett., 13 (2000), 53-56.

[3] G.Bonanno, A minimax inequality and its applications to ordinary differential equations, J. Math. Anal. Appl., to appear.

[4] G.Bonanno, Multiple solutions for a Neumann boundary value problem, J. Convex Nonlin. Anal., to appear.

[5] G.Bonanno and P.Candito, Three solutions to a Neumann problem for elliptic equations involving the p-Laplacian, Arch. Math. (Basel), to appear.

[6] G.Bonanno and R.Livrea, Multiplicity theorems for the Dirichlet problem involving the p-Laplacian, preprint.

[7] G.Bonanno, Some remarks on a three critical points theorem, preprint.

[8] P.Candito, Existence of three solutions for a nonautonomous two point boundary value problem, J. Math. Anal. Appl. 252 (2000), 532-537.

[9] H.Dang, K.Schmitt and R.Shivaji, On the number of solutions of boundary value problems involving the p-laplacian, Electron. J. Differ. Equ. 1996 (1996), 1-9.

[10] J.Henderson and H.B.Thompson, Existence of multiple solutions for second order boundary value problems, J. Differential Equations 166 (2000), 443-454.

[11] R.Livrea, Existence of three solutions for a quasilinear two point boundary value problem, Arch. Math. (Basel), to appear.

[12] S.A.Marano and D.Motreanu, On a three critical points theorem for non-differentiable functions and applications to nonlinear boundary value problems, Nonlinear Anal., 48 (2002), 37-52.

[13] P.Pucci and J.Serrin, A mountain pass theorem, J. Differential Equations 60 (1985), 142-149.

[14] B.Ricceri, A general variational principle and some of its applications, J. Comput. Appl. Math. 113 (2000), 401-410.

[15] B.Ricceri, On a three critical points theorem, Arch. Math. (Basel), 75 (2000), 220-226.

[16] B.Ricceri, Three solutions for a Neumann problem, preprint.