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

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

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

(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, 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 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, 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 such that

for all

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

and

Then, one has

(a) for every r > infX Φ and every , 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 , 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

(c) If δ < +∞ then, for each , 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 , ,... and , i.e., , equipped with the norm

where

(2)

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

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 .

For all γ > 0 we define

(3)

3 Main results

We state our main result as follows:

Theorem 3.1. Assume that

(A1)

where (see (3)).

Then, for each

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

and

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

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

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 . Now, we want to show that

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

Put for all m ∈ ℕ. Since

for each for 1 ≤ i n, we have

(4)

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

Hence, an easy computation shows that whenever u = (u1, ..., un) ∈ Φ-1(] - ∞, rm]). Hence, one has

Therefore, since from Assumption (A1) one has

we deduce

(5)

Assumption (A1) along with (5), implies

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 such that as m

and

(6)

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

Then, for all m ∈ ℕ,

Now, if

we fix . From (6) there exists τε such that

therefore

and by the choice of ε, one has

If

let us consider . From (6) there exists τK such that

therefore

and by the choice of K, one has

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

and we have the conclusion.   □

Here, we give a consequence of Theorem 3.1.

Corollary 3.2. Assume that

(A2)

(A3)

Then, the system

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)

where (see (3)).

Then, for each

the system

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

for every n ≥ 1. Define the function

(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

So

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

Then

and hence

Finally

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

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.

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