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

For all author emails, please log on.

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

- Δ p i u i + a i ( x ) u i p i - 2 u = λ F u i ( x , u 1 , , u n ) in Ω , u i ν = 0 on Ω (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, Δ p i u i = div ( u i p i - 2 u i ) 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 x Ω , F t i 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, F t i 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 u = u 1 , . . . , u n W 1 , p 1 Ω × . . . × W 1 , p n Ω , such that

Ω i = 1 n u i ( x ) p i - 2 u i ( x ) v i ( x ) + a i ( x ) u i ( x ) p i - 2 u i ( x ) v i ( x ) d x - λ Ω i = 1 n F u i ( x , u 1 ( x ) , . . . u n ( x ) ) v i ( x ) d x = 0

for all v = v 1 , . . . , v n W 1 , p 1 Ω × . . . × W 1 , p n Ω .

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

φ ( r ) : = inf u Φ - 1 - , r sup v Φ - 1 - , r Ψ ( v ) - Ψ ( u ) r - Φ ( u )

and

γ : = lim inf r + φ ( r ) , δ : = lim inf r ( inf X Φ ) + φ ( r ) .

Then, one has

(a) for every r > infX Φ and every λ 0 , 1 φ ( r ) , 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 λ 0 , 1 γ , 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

lim n + Φ ( u n ) = + .

(c) If δ < +∞ then, for each λ 0 , 1 δ , 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 W 1 , p 1 ( Ω ) , W 1 , p 2 ( Ω ) ,... and W 1 , p n ( Ω ) , i.e., X = i = 1 n W 1 , p i ( Ω ) , equipped with the norm

u 1 , u 2 , , u n = i = 1 n u i p i ,

where

u i p i = Ω u i ( x ) p i + a i ( x ) u i ( x ) p i d x 1 p i , i = 1 , , n . C = max sup u i W 1 , p i ( Ω ) \ { 0 } sup x Ω u ( x ) p i u i p i p i ; i = 1 , , n . (2)

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

sup u i W 1 , p i ( Ω ) \ { 0 } sup x Ω u i ( x ) u i p i 2 p i - 1 p i max 1 a i 1 1 p i ; diam ( Ω ) N 1 p i p i - 1 p i - N m ( Ω ) p i - 1 p i a i a i 1

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 p ¯ = min { p i ; 1 i n } .

For all γ > 0 we define

K ( γ ) = ( t 1 , , t n ) n : i = 1 n t i γ . (3)

### 3 Main results

We state our main result as follows:

Theorem 3.1. Assume that

(A1)

lim inf ξ + Ω sup ( t 1 , . . . , t n ) K ( ξ ) F ( x , t 1 , , t n ) d x ξ p - < i = 1 n ( p i C ) 1 p i p - lim sup ( t 1 , , t n ) ( t 1 , , t n ) + n Ω F ( x , t 1 , , t n ) d x i = 1 n a i 1 t i p i p i

where K ( ξ ) = { ( t 1 , , t n ) | i = 1 n t i ξ } (see (3)).

Then, for each

λ Λ : = 1 lim sup ( t 1 , , t n ) ( t 1 , , t n ) + n Ω F ( x , t 1 , , t n ) d x i = 1 n | | a i | | 1 | t i | p i , i = 1 n ( p i C ) 1 p i p - lim inf ξ + Ω sup ( t 1 , . . . , t n ) K ( ξ ) F ( x , t 1 , , t n ) d x ξ p -

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

Φ ( u ) = i = 1 n u i p i p i p i

and

Ψ ( u ) = Ω F ( x , u 1 ( x ) , , u n ( x ) ) d x .

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

Ψ ( u ) ( v ) = Ω i = 1 n F u i ( x , u 1 ( x ) , , u n ( x ) ) v i ( x ) d x

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

Φ ( u 1 , , u n ) ( v 1 , , v n ) Ω i = 1 n u i ( x ) p i - 2 u i ( x ) v i ( x ) + a i ( x ) u i ( x ) p i - 2 u i ( x ) v i ( x ) d x

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 I λ ( u 1 , , u n ) = 0 . Now, we want to show that

γ < + .

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

lim m Ω sup ( t 1 , , t n ) K ( ξ m ) F ( x , t 1 , , t n ) d x ξ m p - = lim inf ξ + Ω sup ( t 1 , , t n ) K ( ξ ) F ( x , t 1 , , t n ) d x ξ p - .

Put r m = ξ m p - i = 1 n ( p i C ) 1 p i p - for all m ∈ ℕ. Since

sup x Ω u i ( x ) p i C u i p i p i

for each u i W 1 , p i ( Ω ) for 1 ≤ i n, we have

sup x Ω i = 1 n u i ( x ) p i p i C i = 1 n u i p i p i p i . (4)

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

Φ - 1 - , r u X ; sup i = 1 n u i ( x ) p i p i C r for each x Ω .

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

φ ( r m ) = inf u Φ - 1 - , r m ( sup v Φ - 1 - , r m Ψ ( v ) ) - Φ ( u ) r m - Φ ( u ) sup v Φ - 1 - , r m Ψ ( v ) r m Ω sup ( t 1 , , t n ) K ( ξ m ) F ( x , t 1 , , t n ) d x ξ m p - i = 1 n ( p i C ) 1 p i p - .

Therefore, since from Assumption (A1) one has

lim inf ξ + Ω sup ( t 1 , , t n ) K ( ξ ) F ( x , t 1 , , t n ) d x ξ p - < ,

we deduce

γ lim inf m + φ ( r m ) i = 1 n ( p i C ) 1 p i p - lim inf ξ + Ω sup ( t 1 , , t n ) K ( ξ ) F ( x , t 1 , , t n ) d x ξ p - < + . (5)

Assumption (A1) along with (5), implies

Λ 0 , 1 γ .

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 { d i , m } i = 1 n such that i = 1 n d i , m 2 + as m

and

lim m + Ω F ( x , d 1 , m , , d n , m ) d x i = 1 n d i , m p i p i = lim sup ( t 1 , , t n ) ( t 1 , , t n ) + n Ω F ( x , t 1 , , t n ) d x i = 1 n a i 1 t i p i p i . (6)

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

Φ ( w m ) = i = 1 n d i , m p i a i 1 p i .

Then, for all m ∈ ℕ,

I λ ( w m ) = Φ ( w m ) - λ Ψ ( w m ) = i = 1 n d i , m p i a i 1 p i - λ Ω F ( x , d 1 , m , , d n , m ) d x .

Now, if

lim sup ( t 1 , , t n ) ( t 1 , , t n ) + n Ω F ( x , t 1 , , t n ) d x i = 1 n a i 1 | t i | p i p i < ,

we fix ε 1 λ lim sup ( t 1 , , t n ) ( t 1 , , t n ) + n Ω F ( x , t 1 , , t n ) d x i = 1 n a i 1 t i p i p i , 1 . From (6) there exists τε such that

Ω F ( x , d 1 , m , , d n , m ) d x > ε lim sup ( t 1 , , t n ) ( t 1 , , t n ) + n Ω F ( x , t 1 , , t n ) d x i = 1 n a i 1 t i p i p i i = 1 n d i , m p i a i 1 p i m > τ ε ,

therefore

I λ ( w m ) 1 - λ ε lim sup ( t 1 , , t n ) ( t 1 , , t n ) + n Ω F ( x , t 1 , , t n ) d x i = 1 n a i 1 t i p i p i i = 1 n d i , m p i a i 1 p i m > τ ε ,

and by the choice of ε, one has

lim m + [ Φ ( w m ) - λ Ψ ( w m ) ] = - .

If

lim sup ξ + Ω F ( x , t 1 , , t n ) d x i = 1 n a i 1 t i p i p i = ,

let us consider K > 1 λ . From (6) there exists τK such that

Ω F ( x , d 1 , m , , d n , m ) d x > K i = 1 n d i , m p i a i 1 p i m > τ K ,

therefore

I λ ( w m ) ( 1 - λ K ) i = 1 n d i , m p i a i 1 p i m > τ K ,

and by the choice of K, one has

lim m + [ Φ ( w m ) - λ Ψ ( w m ) ] = - .

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

lim m ( u 1 m , , u n m ) = + ,

and we have the conclusion.   □

Here, we give a consequence of Theorem 3.1.

Corollary 3.2. Assume that

(A2) lim inf ξ + Ω sup ( t 1 , , t n ) K ( ξ ) F ( x , t 1 , , t n ) d x ξ p - < i = 1 n ( p i C ) 1 p i p - ;

(A3) lim sup ( t 1 , , t n ) ( t 1 , , t n ) + n Ω F ( x , t 1 , , t n ) d x i = 1 n a i 1 t i p i p i > 1 .

Then, the system

- Δ p i u i + a i ( x ) u i p i - 2 u = F u i ( x , u 1 , , u n ) i n Ω , u i ν = 0 o n   Ω

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)

lim inf ξ + sup ( t 1 , , t n ) K ( ξ ) F ( t 1 , , t n ) ξ p - < i = 1 n ( p i C ) 1 p i p - lim sup ( t 1 , , t n ) ( t 1 , , t n ) + n F ( t 1 , , t n ) i = 1 n a i 1 t i p i p i

where K ( ξ ) = { ( t 1 , , t n ) | i = 1 n t i ξ } (see (3)).

Then, for each

λ Λ : = 1 F ( t 1 , , t n ) lim sup ( t 1 , , t n ) ( t 1 , , t n ) + n i = 1 n a i 1 t i p i , i = 1 n ( p i C ) 1 p i p - lim inf ξ + sup ( t 1 , , t n ) K ( ξ ) F ( t 1 , , t n ) ξ p -

the system

- Δ p i u i + a i ( x ) u i p i - 2 u = λ F u i ( u 1 , , u n ) i n Ω , u i ν = 0 o n   Ω

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 n : = 2 , a n + 1 : = n ! ( a n ) 5 4 + 2

for every n ≥ 1. Define the function

F ( t 1 , t 2 ) = ( a n + 1 ) 5 e - 1 1 - [ ( t 1 - a n + 1 ) 2 + ( t 2 - a n + 1 ) 2 ] ( t 1 , t 2 ) n 1 B ( ( a n + 1 , a n + 1 ) , 1 ) , 0 otherwise (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

lim sup n + F ( a n + 1 , a n + 1 ) a n + 1 3 3 + a n + 1 4 4 = +

So

lim sup ( t 1 , t 2 ) ( + , + ) F ( t 1 , t 2 ) t 1 3 3 + t 2 4 4 = +

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

sup ( t 1 , t 2 ) K ( y n ) F ( t 1 , t 2 ) = a n 5 n

Then

lim n sup ( t 1 , t 2 ) K ( y n ) F ( t 1 , t 2 ) ( a n + 1 - 1 ) 3 = 0 ,

and hence

lim inf ξ sup ( t 1 , t 2 ) K ( ξ ) F ( t 1 , t 2 ) ξ 3 = 0 .

Finally

0 = lim inf ξ + sup ( t 1 , t 2 ) K ( ξ ) F ( t 1 , t 2 ) ξ 3 < ( ( 3 C ) 1 3 + ( 4 C ) 1 4 ) 3 lim sup ( t 1 , t 2 ) ( + , + ) ( t 1 , t 2 ) + n F ( t 1 , t 2 ) t 1 3 3 + t 2 4 4 = + .

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

- Δ 3 u + u u = λ f ( u , v ) in Ω , - Δ 4 v + v 2 g = λ g ( u , v ) in Ω , u ν = v ν = 0 on Ω

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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)