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

for i = 1, ..., n, where Ω ⊂ ℝ^N (N ≥ 1) is a non-empty bounded open set with a smooth boundary ∂Ω, p_i > N for i = 1, ..., n, Δ p i u i = div ( ∇ u i p i - 2 ∇ u i ) is the p_i-Laplacian operator, a_i ∈ L^∞ (Ω) with ess inf_Ω a_i > 0 for i = 1, ..., n, λ > 0, and F: Ω × ℝⁿ → ℝ is a function such that the mapping (t₁, t₂,..., t_n) → F (x, t₁, t₂,..., t_n) 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 t_i.

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 1and its variants 23 we refer the reader to the articles 45678910111213141516.

For other basic definitions and notations we refer the reader to the articles 171819202122. 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.

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 > inf_X 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 > inf_X Φ and every λ ∈ 0 , 1 φ ( r ) , the restriction of the functional I_λ = Φ - λΨ to Φ^-1(] - ∞, r[) admits a global minimum, which is a critical point (local minimum) of I_λ in X.

(b) If γ < +∞ then, for each λ ∈ 0 , 1 γ , the following alternative holds:

either

(b₁) I_λ possesses a global minimum,

or

(b₂) there is a sequence {u_n} of critical points (local minima) of I_λ 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 I_λ,

or

(c₂) there is a sequence {u_n} of pairwise distinct critical points (local minima) of I_λ 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 .

Since p_i > 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, ||·||_∞ = sup_x∈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 ) ∈ ℝ n : ∑ i = 1 n t i ≤ γ .

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 ) ∈ ℝ + 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 ) ∈ ℝ + 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, ..., u_n) ∈ 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, ..., v_n) ∈ 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, ..., v_n) ∈ X. Furthermore, (Φ')^-1: X* → X exists and is continuous.

Put I_λ: = 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 {ξ_m} be a real sequence such that ξ_m → +∞ 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 .

for each u = (u₁, u₂, ..., u_n) ∈ 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, ..., u_n) ∈ Φ^-1(] - ∞, r_m]). 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 - < + ∞ .

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 I_λ has a global minimum or there exists a sequence {u_m} where u_m = (u_1m, ..., u_nm) of weak solutions of the system (1) such that lim_m→∞ ||(u_1m, ..., u_nm)|| = +∞.

Now fix λ ∈ Λ and let us verify that the functional I_λ 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 ) ∈ ℝ + n ∫ Ω F ( x , t 1 , … , t n ) d x ∑ i = 1 n a i 1 t i p i p i .

For all m ∈ ℕ define w_m(x) = (d_{1, m}, ..., d_{n, m}). For any fixed m ∈ ℕ, w_m ∈ 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 ) ∈ ℝ + 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 ) ∈ ℝ + 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 τ_ε such that

∫ Ω F ( x , d 1 , m , … , d n , m ) d x > ε lim sup ( t 1 , … , t n ) → ∞ ( t 1 , … , t n ) ∈ ℝ + 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 ) ∈ ℝ + 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 τ_K 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 I_λ admits a sequence {u_m = (u_1m, ..., u_nm)} ⊂ 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 ) ∈ ℝ + 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 ) ∈ ℝ + 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 ) ∈ ℝ + 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, ..., u_n) = F (u₁, ..., u_n) for all x ∈ Ω and (u₁, ..., u_n) ∈ ℝⁿ. The conclusion follows from Theorem 3.1. □

Remark 3.1. We observe in Theorem 3.1 we can replace ξ → +∞ and (t₁, ..., t_n) → (+∞, ..., +∞) with ξ → 0⁺ (t₁, ..., t_n) → (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

where B((a_n+1, a_n+1), 1)) be the open unit ball of center (a_n+1, a_n+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((a_n+1, a_n+1), 1) attains its maximum in (a_n+1, a_n+1) and F (a_n+1, a_n+1) = (a_n+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 y_n = a_n+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 ∈ 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 ) ∈ ℝ + 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, +∞[.

The authors declare that they have no competing interests.

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.