Open Access Research

Infinitely many solutions for class of Neumann quasilinear elliptic systems

Davood Maghsoodi Shoorabi1* and Ghasem Alizadeh Afrouzi2

Author Affiliations

1 Department of Mathematics, Science and Research Branch, Islamic Azad University (IAU), Tehran, Iran

2 Department of Mathematics, Faculty of Basic Sciences, University of Mazandaran, 47416-1467 Babolsar, Iran

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Boundary Value Problems 2012, 2012:54  doi:10.1186/1687-2770-2012-54


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2012/1/54


Received:30 January 2012
Accepted:6 May 2012
Published:6 May 2012

© 2012 Shoorabi and Afrouzi; licensee Springer.

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We investigate the existence of infinitely many weak solutions for a class of Neumann quasilinear elliptic systems driven by a (p1, ..., pn)-Laplacian operator. The technical approach is fully based on a recent three critical points theorem.

AMS subject classification: 35J65; 34A15.

Keywords:
infinitely many solutions; Neumann system; critical point theory; variational methods

1 Introduction

The purpose of this article is to establish the existence of infinitely many weak solutions for the following Neumann quasilinear elliptic system

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M1">View MathML</a>

(1)

for i = 1, ..., n, where Ω ⊂ ℝN (N ≥ 1) is a non-empty bounded open set with a smooth boundary ∂Ω, pi > N for i = 1, ..., n, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M2">View MathML</a> is the pi-Laplacian operator, ai L(Ω) with ess infΩ ai > 0 for i = 1, ..., n, λ > 0, and F: Ω × ℝn → ℝ is a function such that the mapping (t1, t2,..., tn) → F (x, t1, t2,..., tn) is in C1 in ℝn for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M3">View MathML</a> is continuous in Ω × ℝn for i = 1,..., n, and F (x, 0,..., 0) = 0 for all x ∈ Ω and ν is the outward unit normal to ∂Ω. Here, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M4">View MathML</a> denotes the partial derivative of F with respect to ti.

Precisely, under appropriate hypotheses on the behavior of the nonlinear term F at infinity, the existence of an interval Λ such that, for each λ ∈ Λ, the system (1) admits a sequence of pairwise distinct weak solutions is proved; (see Theorem 3.1). We use a variational argument due to Ricceri which provides certain alternatives in order to find sequences of distinct critical points of parameter-depending functionals. We emphasize that no symmetry assumption is required on the nonlinear term F (thus, the symmetry version of the Mountain Pass theorem cannot be applied). Instead of such a symmetry, we assume a suitable oscillatory behavior at infinity on the function F.

We recall that a weak solution of the system (1) is any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M5">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M6">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M7">View MathML</a>

For a discussion about the existence of infinitely many solutions for differential equations, using Ricceri's variational principle [1]and its variants [2,3] we refer the reader to the articles [4-16].

For other basic definitions and notations we refer the reader to the articles [17-22]. Here, our motivation comes from the recent article [8]. We point out that strategy of the proof of the main result and Example 3.1 are strictly related to the results and example contained in [8].

2 Preliminaries

Our main tool to ensure the existence of infinitely many classical solutions for Dirichlet quasilinear two-point boundary value systems is the celebrated Ricceri's variational principle [[1], Theorem 2.5] that we now recall as follows:

Theorem 2.1. 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. For every r > infX Φ, let us put

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M8">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M9">View MathML</a>

Then, one has

(a) for every r > infX Φ and every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M10">View MathML</a>, the restriction of the functional Iλ = Φ - λΨ to Φ-1(] - ∞, r[) admits a global minimum, which is a critical point (local minimum) of Iλ in X.

(b) If γ < +∞ then, for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M11">View MathML</a>, the following alternative holds:

either

(b1) Iλ possesses a global minimum,

or

(b2) there is a sequence {un} of critical points (local minima) of Iλ such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M12">View MathML</a>

(c) If δ < +∞ then, for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M13">View MathML</a>, the following alternative holds:

either

(c1) there is a global minimum of Φ which is a local minimum of Iλ,

or

(c2) there is a sequence {un} of pairwise distinct critical points (local minima) of Iλ that converges weakly to a global minimum of Φ.

We let X be the Cartesian product of n Sobolev spaces <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M14">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M15">View MathML</a>,... and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M16">View MathML</a>, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M17">View MathML</a>, equipped with the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M18">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M19">View MathML</a>

(2)

Since pi > N for 1 ≤ i ≤ n, one has C < +∞. In addition, if Ω is convex, it is known [23] that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M20">View MathML</a>

for 1 ≤ i ≤ n, where ||·||1 = ∫Ω|·(x)| dx, ||·||= supx∈Ω |·(x)| and m(Ω) is the Lebesgue measure of the set Ω, and equality occurs when Ω is a ball.

In the sequel, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M21">View MathML</a>.

For all γ > 0 we define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M22">View MathML</a>

(3)

3 Main results

We state our main result as follows:

Theorem 3.1. Assume that

(A1)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M23">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M24">View MathML</a>(see (3)).

Then, for each

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M25">View MathML</a>

the system (1) has an unbounded sequence of weak solutions in X.

Proof. Define the functionals Φ, Ψ: X → ℝ for each u = (u1, ..., un) ∈ X, as follows

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M26">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M27">View MathML</a>

It is well known that Ψ is a Gâteaux differentiable functional and sequentially weakly lower semicontinuous whose Gâteaux derivative at the point u X is the functional Ψ'(u) ∈ X*, given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M28">View MathML</a>

for every v = (v1, ..., vn) ∈ X, and Ψ': X X* is a compact operator. Moreover, Φ is a sequentially weakly lower semicontinuous and Gâteaux differentiable functional whose Gâteaux derivative at the point u X is the functional Φ' (u) ∈ X*, given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M29">View MathML</a>

for every v = (v1, ..., vn) ∈ X. Furthermore, (Φ')-1: X* X exists and is continuous.

Put Iλ: = Φ - λΨ. Clearly, the weak solutions of the system (1) are exactly the solutions of the equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M30">View MathML</a>. Now, we want to show that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M31">View MathML</a>

Let {ξm} be a real sequence such that ξm → +∞ as m → ∞ and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M32">View MathML</a>

Put <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M33">View MathML</a> for all m ∈ ℕ. Since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M34">View MathML</a>

for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M35">View MathML</a> for 1 ≤ i n, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M36">View MathML</a>

(4)

for each u = (u1, u2, ..., un) ∈ X. This, for each r > 0, together with (4), ensures that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M37">View MathML</a>

Hence, an easy computation shows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M38">View MathML</a> whenever u = (u1, ..., un) ∈ Φ-1(] - ∞, rm]). Hence, one has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M39">View MathML</a>

Therefore, since from Assumption (A1) one has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M40">View MathML</a>

we deduce

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M41">View MathML</a>

(5)

Assumption (A1) along with (5), implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M42">View MathML</a>

Fix λ ∈ Λ. The inequality (5) concludes that the condition (b) of Theorem 2.1 can be applied and either Iλ has a global minimum or there exists a sequence {um} where um = (u1m, ..., unm) of weak solutions of the system (1) such that limm→∞ ||(u1m, ..., unm)|| = +.

Now fix λ ∈ Λ and let us verify that the functional Iλ is unbounded from below. Arguing as in [8], consider n positive real sequences <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M43">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M44">View MathML</a> as m

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M45">View MathML</a>

(6)

For all m ∈ ℕ define wm(x) = (d1, m, ..., dn, m). For any fixed m ∈ ℕ, wm X and, in particular, one has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M46">View MathML</a>

Then, for all m ∈ ℕ,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M47">View MathML</a>

Now, if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M48">View MathML</a>

we fix <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M49">View MathML</a>. From (6) there exists τε such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M50">View MathML</a>

therefore

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M51">View MathML</a>

and by the choice of ε, one has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M52">View MathML</a>

If

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M53">View MathML</a>

let us consider <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M54">View MathML</a>. From (6) there exists τK such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M55">View MathML</a>

therefore

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M56">View MathML</a>

and by the choice of K, one has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M57">View MathML</a>

Hence, our claim is proved. Since all assumptions of Theorem 2.1 are satisfied, the functional Iλ admits a sequence {um = (u1m, ..., unm)} ⊂ X of critical points such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M58">View MathML</a>

and we have the conclusion.   □

Here, we give a consequence of Theorem 3.1.

Corollary 3.2. Assume that

(A2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M59">View MathML</a>

(A3) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M60">View MathML</a>

Then, the system

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M61">View MathML</a>

for 1 ≤ i ≤ n, has an unbounded sequence of classical solutions in X.

Now, we want to present the analogous version of the main result (Theorem 3.1) in the autonomous case.

Theorem 3.3. Assume that

(A4)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M62">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M63">View MathML</a> (see (3)).

Then, for each

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M64">View MathML</a>

the system

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M65">View MathML</a>

has an unbounded sequence of weak solutions in X.

Proof. Set F (x, u1, ..., un) = F (u1, ..., un) for all x ∈ Ω and (u1, ..., un) ∈ ℝn. The conclusion follows from Theorem 3.1. □

Remark 3.1. We observe in Theorem 3.1 we can replace ξ → +∞ and (t1, ..., tn) → (+, ..., +∞) with ξ → 0+ (t1, ..., tn) → (0+, ..., 0+), respectively, that by the same way as in the proof of Theorem 3.1 but using conclusion (c) of Theorem 2.1 instead of (b), the system (1) has a sequence of weak solutions, which strongly converges to 0 in X.

Finally, we give an example to illustrate the result.

Example 3.1. Let Ω ⊂ ℝ2 be a non-empty bounded open set with a smooth boundary ϑΩ and consider the increasing sequence of positive real numbers given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M66">View MathML</a>

for every n ≥ 1. Define the function

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M67">View MathML</a>

(7)

where B((an+1, an+1), 1)) be the open unit ball of center (an+1, an+1). We observe that the function F is non-negative, F (0, 0) = 0, and F C1(ℝ2). We will denote by f and g, respectively, the partial derivative of F respect to t1 and t2. For every n ∈ ℕ, the restriction F on B((an+1, an+1), 1) attains its maximum in (an+1, an+1) and F (an+1, an+1) = (an+1)5,

then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M68">View MathML</a>

So

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M69">View MathML</a>

On the other by setting yn = an+1 - 1 for every n ∈ ℕ, one has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M70">View MathML</a>

Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M71">View MathML</a>

and hence

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M72">View MathML</a>

Finally

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M73">View MathML</a>

So, since all assumptions of Theorem 3.3 is applicable to the system

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/54/mathml/M74">View MathML</a>

for every λ ∈ [0, +[.

Competing interests

The authors declare that they have no competing interests.

Authors' contributions

DMS has presented the main purpose of the article and has used GAA contribution due to reaching to conclusions. All authors read and approved the final manuscript.

References

  1. Ricceri, B: A general variational principle and some of its applications. J Comput Appl Math. 113, 401–410 (2000). Publisher Full Text OpenURL

  2. Bonanno, G, Molica Bisci, G: Infinitely many solutions for a boundary value problem with discontinuous nonlinearities. Bound Value Probl. 2009, 1–20 (2009)

  3. Marano, S, Motreanu, D: Infinitely many critical points of non-differentiable functions and applications to the Neumann-type problem involving the p-Laplacian. J Diff Equ. 182, 108–120 (2002). Publisher Full Text OpenURL

  4. Bonanno, G, D'Aguì, G: On the Neumann problem for elliptic equations involving the p-Laplacian. J Math Anal Appl. 358, 223–228 (2009). Publisher Full Text OpenURL

  5. Bonanno, G, Di Bella, B: Infinitely many solutions for a fourth-order elastic beam equation. Nonlinear Diff Equ Appl NoDEA. 18, 357–368 (2011). Publisher Full Text OpenURL

  6. Bonanno, G, Molica Bisci, G: A remark on perturbed elliptic Neumann problems. Studia Univ "Babeş-Bolyai", Mathematica. LV(4), (2010)

  7. Bonanno, G, Molica Bisci, G: Infinitely many solutions for a Dirichlet problem involving the p-Laplacian. Proc Royal Soc Edinburgh. 140A, 737–752 (2010)

  8. Bonanno, G, Molica Bisci, G, O'Regan, D: Infinitely many weak solutions for a class of quasilinear elliptic systems. Math Comput Model. 52, 152–160 (2010). Publisher Full Text OpenURL

  9. Bonanno, G, Molica Bisci, G, Rădulescu, V: Infinitely many solutions for a class of nonlinear eigenvalue problems in Orlicz-Sobolev spaces. C R Acad Sci Paris, Ser I. 349, 263–268 (2011). Publisher Full Text OpenURL

  10. Candito, P: Infinitely many solutions to the Neumann problem for elliptic equations involving the p-Laplacian and with discontinuous nonlinearities. Proc Edin Math Soc. 45, 397–409 (2002)

  11. Candito, P, Livrea, R: Infinitely many solutions for a nonlinear Navier boundary value problem involving the p-biharmonic. Studia Univ "Babeş-Bolyai", Mathematica. LV(4), (2010)

  12. Dai, G: Infinitely many solutions for a Neumann-type differential inclusion problem involving the p(x)-Laplacian. Nonlinear Anal. 70, 2297–2305 (2009). Publisher Full Text OpenURL

  13. Fan, X, Ji, C: Existence of infinitely many solutions for a Neumann problem involving the p(x)-Laplacian. J Math Anal Appl. 334, 248–260 (2007). Publisher Full Text OpenURL

  14. Kristály, A: Infinitely many solutions for a differential inclusion problem in ℝN. J Diff Equ. 220, 511–530 (2006). Publisher Full Text OpenURL

  15. Li, C: The existence of infinitely many solutions of a class of nonlinear elliptic equations with a Neumann boundary conditions for both resonance and oscillation problems. Nonlinear Anal. 54, 431–443 (2003). Publisher Full Text OpenURL

  16. Ricceri, B: Infinitely many solutions of the Neumann problem for elliptic equations involving the p-Laplacian. Bull Lond Math Soc. 33(3), 331–340 (2001). Publisher Full Text OpenURL

  17. Afrouzi, GA, Heidarkhani, S: Existence of three solutions for a class of Dirichlet quasi-linear elliptic systems involving the (p1, ..., pn)-Laplacian. Nonlinear Anal. 70, 135–143 (2009). Publisher Full Text OpenURL

  18. Afrouzi, GA, Heidarkhani, S, O'Regan, D: Three solutions to a class of Neumann doubly eigenvalue elliptic systems driven by a (p1, ..., pn)-Laplacian. Bull Korean Math Soc. 47(6), 1235–1250 (2010). Publisher Full Text OpenURL

  19. Bonanno, G, Heidarkhani, S, O'Regan, D: Multiple solutions for a class of Dirichlet quasilinear elliptic systems driven by a (p, q)-Laplacian operator. Dyn Syst Appl. 20, 89–100 (2011)

  20. Heidarkhani, S, Tian, Y: Multiplicity results for a class of gradient systems depending on two parameters. Nonlinear Anal. 73, 547–554 (2010). Publisher Full Text OpenURL

  21. Heidarkhani, S, Tian, Y: Three solutions for a class of gradient Kirchhoff-type systems depending on two parameters. Dyn Syst Appl. 20, 551–562 (2011)

  22. Zeidler, E: Nonlinear Functional Analysis and its Applications. Springer, New York (1985)

  23. Bonanno, G, Candito, P: Three solutions to a Neumann problem for elliptic equations involving the p-Laplacian. Arch Math (Basel). 80, 424–429 (2003)