In this article, we consider the problem of the type

- Δ p ( x ) u + e p ( x ) u p ( x ) - 2 u = λ F u ( x , u , v ) + μ G u ( x , u , v ) , x ∈ Ω , - Δ q ( x ) v + e p ( x ) v q ( x ) - 2 v = λ F v ( x , u , v ) + μ G v ( x , u , v ) , x ∈ Ω , ∂ u ∂ v = ∂ v ∂ v = 0 x ∈ ∂ Ω ,

where Ω ⊂ R^N(N ≥ 2) is a bounded domain with boundary of class C¹. ν is the outer unit normal to ∂Ω, λ, μ ≥ 0 are real numbers. p ( x ) , q ( x ) ∈ C 0 Ω ̄ with N < p − : = inf x ∈ Ω ¯ p ( x ) ≤ p + : = sup x ∈ Ω ¯ p ( x ) , , N < q^- ≤ q⁺, F : Ω × R × R → R is a function such that F(·, s, t) is measurable in Ω for all (s, t) ∈ R × R and F(x, ·, ·) is C¹ in R × R for a.e. x ∈ Ω, F_s denotes the partial derivative of F with respect to s. We assume G(x,s,t) and e_p(x),e_q(x) satisfy the following conditions:

(G) G : Ω × R × R → R is a Carathéodory function, sup_{{|s|≤θ,|t|≤ϑ}} |G(·,s,t)| ∈ L¹(Ω) for all θ, ϑ > 0;

(E) e_p(x),e_q(x) ∈ L^∞(Ω) and ess inf_Ω e_p(x), ess inf_Ω e_q(x) > 0, we denote ∥e_p∥₁ = ∫_Ω e_p(x)dx and ∥e_q∥₁ = ∫_Ωe_q(x)dx.

It is well known that the operator -Δ_p(x) = -div(|∇u|^p(x)-2∇u) is called p(x)-Laplacian and the corresponding problem is called a variable exponent elliptic systems. The study of differential equations and variational problems with nonstandard p(x)-growth conditions has been attracting attention of many authors in the last two decades. It arises from nonlinear elasticity theory, electro-rheological fluids, etc. see 12, many results have been obtained on this kind of problems, for example 3456789. For the special case, p(x) ≡ p(a constant), (1.1) becomes the well known p-Laplacian problem. There have been many papers on this class of problems, see 10111213141516171819 and the reference therein.

Recently, many papers have appeared in which the technical approach adopted is based on the three critical points theorem obtained by Ricceri 16. We cite papers 20212223, where the authors established the existence of at least three weak solutions to the problems with Dirichlet or Neumann boundary value conditions. Li and Tang in 24 obtained the existence of at least three weak solutions to problem (1) when p(x) ≡ p with Dirichlet boundary value conditions. El Manouni and Kbiri Alaoui 25 obtained the existence of at least three solutions of system (1) when p(x) ≡ p in Ω by the three critical points theorem obtained by Ricceri 26.

The main purpose of the present paper is to prove the existence of at least three solutions of problem (1). We study problem (1) by using the three critical points theorem by Ricceri 26 too. On the basis of 27, we state an equivalent formulation of the three critical points theorem in 26 as follows.

Theorem 1. Let X be a reflexive real Banach space, Φ : X → R a continuously Gâteaux differentiable and sequentially weakly lower semicontinuous C¹ functional, bounded on each bounded subset of X, whose Gâteaux derivative admits a continuous inverse on X*; Ψ : X → R a C¹ functional with compact Gâteaux derivative. Assume that

(i) lim_{∥u∥→∞}(Φ(u) + λ Ψ(u)) = ∞ for all λ > 0; and there are r ∈ R and u₀, u₁ ∈ X such that:

(ii) Φ(u₀) < r < Φ(u₁);

(iii) inf u ∈ Φ − 1 ( ( − ∞ , r ] ) Ψ ( u ) > ( Φ ( u 1 ) − r ) Ψ ( u 0 ) + ( r − Φ ( u 0 ) ) Ψ ( u 1 ) Φ ( u 1 ) − Φ ( u 0 ) . .

Then there exists a non-empty open set Λ ⊆ [0, ∞) and a positive real number ρ with the following property: for each λ ∈ Λ and every C¹ functional J : X → R with compact Gâteaux derivative, there exists σ > 0 such that for each μ ∈ [0, σ], the equation

Φ ′ ( u ) + λ Ψ ′ ( u ) + μ J ′ ( u ) = 0

has at least three solutions in X whose norms are less than ρ.

The paper is organized as follows. In section 2, we recall some facts that will be needed in the paper. In section 3, we establish our main result.

In order to deal with p(x)-Laplacian problem, we need some theories on spaces L^p(x)(Ω), W^1,p(x)(Ω) and properties of p(x)-Laplacian which we will use later (see 152829).

We denote

L p ( x ) ( Ω ) = u | u is a measurable real - valued function on Ω , ∫ Ω u ( x ) p ( x ) d x < ∞ .

We can introduce a norm on L^p(x)(Ω) by

u p ( x ) = inf λ > 0 | ∫ Ω u ( x ) λ p ( x ) d x ≤ 1 .

and (L^p(x)(Ω), | · |_p(x)) becomes a Banach space, and we call it variable exponent Lebesgue space.

The space W^1,p(x)(Ω) is defined by

W 1 , p ( x ) ( Ω ) = u ∈ L p ( x ) ( Ω ) | ∇ u ∈ L p ( x ) ( Ω ) ,

and it can be equipped with the norm

u p ( x ) = u p ( x ) + ∇ u p ( x ) , ∀ u ∈ W 1 , p ( x ) ( Ω ) ,

and we call it variable exponent Sobolev space. From 5, we know that spaces L^p(x)(Ω) and W^1,p(x)(Ω) are separable, reflexive and uniform convex Banach spaces.

When e_p(x) satisfy (E), we define

L e p ( x ) p ( x ) ( Ω ) = u | u is a measurable real - valued function, ∫ Ω e p ( x ) u ( x ) p ( x ) d x < ∞ ,

with the norm

u ( p ( x ) , e p ( x ) ) = inf λ > 0 | ∫ Ω e p ( x ) u ( x ) λ p ( x ) d x ≤ 1 ,

then L e p ( x ) p ( x ) ( Ω ) is a Banach space. For any u ∈ W^1,p(x)(Ω), define

u e p = inf λ > 0 | ∫ Ω ∇ u ( x ) λ p ( x ) + e p ( x ) u ( x ) λ p ( x ) d x ≤ 1 .

Then it is easy to see that u e p is a norm on W^1,p(x)(Ω) equivalent to ∥u∥_p(x). In the following, we will use ⋅ e p to instead of ∥ · ∥_p(x) on W^1,p(x)(Ω). Similarly, we use ⋅ep to instead of ∥ · ∥_q(x) on W^1,q(x)(Ω).

Proposition 1. (see 15) The conjugate space of L^p(x)(Ω) is L p 0 ( x ) ( Ω ) , where 1 p ( x ) + 1 p 0 ( x ) = 1 . For any u ∈ L^p(x)(Ω) and v ∈ L p 0 ( x ) ( Ω ) , we have

∫ Ω u v d x ≤ 1 p - + 1 p 0 - u p ( x ) v p 0 ( x ) ≤ 2 u p ( x ) v p 0 ( x ) .

Proposition 2. (see 15)If we denote ρ(u) = ∫_Ω |u|^p(x)dx, ∀u ∈ L^p(x)(Ω), then

(i) |u|_p(x) < 1(= 1; > 1) ⇔ ρ (u) < 1(= 1; > 1);

(ii) u p ( x ) > 1 ⇒ u p ( x ) p - ≤ ρ ( u ) ≤ u p ( x ) p + ; u p ( x ) < 1 ⇒ u p ( x ) p + ≤ ρ ( u ) ≤ u p ( x ) p - ;

(iii) |u|_p(x) → 0(∞) ⇔ ρ (u) → 0(∞).

From Proposition 2, the following inequalities hold:

u e p p - ≤ ∫ Ω ∇ u ( x ) p ( x ) + e p ( x ) u ( x ) p ( x ) d x ≤ u e p p + , if u e p ≥ 1 ;

u e p p + ≤ ∫ Ω ∇ u ( x ) p ( x ) + e p ( x ) u ( x ) p ( x ) d x ≤ u e p p - , if u e p ≤ 1 .

Proposition 3.If Ω ⊂ R^N is a bounded domain, then the imbedding W 1 , p ( x ) ( Ω ) ↪ C 0 ( Ω ̄ ) is compact whenever N < p^-.

Proof. It is well know that W 1 , p ( x ) ( Ω ) ↪ W 1 , p - ( Ω ̄ ) is a continuous embedding, and the embedding W 1 , p - ( Ω ) ↪ C 0 ( Ω ̄ ) is compact when N < p^- and Ω is bounded. So we obtain the embedding W1,p(x)(Ω)↪C0(Ω̄) is compact whenever N < p^-.

From now on, we denote X by W^1,p(x)(Ω) × W^1,q(x)(Ω) with the norm

z = u e p + v e p for any z = ( u , v ) ∈ X .

Then X is a separable and reflexive Banach spaces. Naturally, we denote X* by the space (W^1,p(x))*(Ω) × (W^1,q(x))*(Ω), the dual space of X.

From Proposition 3, we know that when p^-,q^- > N, the embedding x ↪ C 0 ( Ω ̄ ) × C 0 ( Ω ̄ ) is compact, there exist a positive constant c such that

z ∞ = u ∞ + v ∞ = sup x ∈ Ω ̄ u ( x ) + sup x ∈ Ω ̄ v ( x ) ≤ c z .

We define Φ, Ψ, J : X → R by

Φ ( z ) = ∫ Ω 1 p ( x ) ∇ u ( x ) p ( x ) + e p ( x ) u ( x ) p ( x ) d x + ∫ Ω 1 q ( x ) ∇ v ( x ) q ( x ) + e p ( x ) v ( x ) q ( x ) d x ,

Ψ ( z ) = - ∫ Ω F ( x , u , v ) d x ,

J ( z ) = ∫ Ω G ( x , u , v ) d x .

Then for any (ζ,η) ∈ X,

Φ ′ ( z ) , ( ζ , η ) = ∫ Ω ∇ u p ( x ) - 2 ∇ u ∇ ζ + e p ( x ) u p ( x ) - 2 u ζ d x + ∫ Ω ∇ v q ( x ) - 2 ∇ v ∇ η + e q ( x ) v q ( x ) - 2 v η d x ∀ z ∈ X , Ψ ′ ( z ) , ( ζ , η ) = ∫ Ω F u ( x , u , v ) ζ d x - ∫ Ω F v ( x , u , v ) η d x , ∀ z ∈ X . J ′ ( z ) , ( ζ , η ) = ∫ Ω G u ( x , u , v ) ζ d x - ∫ Ω G v ( x , u , v ) η d x , ∀ z ∈ X .

We say that z = (u, v) ∈ X is a weak solution of problem (1) if for any (ζ, η) ∈ X

Ψ ′ ( z ) , ( ζ , η ) + λ Ψ ′ ( z ) , ( ζ , η ) + μ ( J ′ ( z ) , ( ζ , η ) ) = 0 .

Thus, we deduce that z ∈ X is a weak solution of (1) if z is a solution of (2). It follows that we can seek for weak solutions of (1) by applying Theorem 1.

We first give the following result.

Lemma 1. If Φ is defined in (6), then (Φ')^-1 : X* → X exists and it is continuous.

Proof. First, we show that Φ' is uniformly monotone. In fact, for any ζ, η ∈ R^N, we have the following inequality (see 30):

ζ p - 2 ζ - η p - 2 η ( ζ - η ) ≥ 1 2 p ζ - η , p ≥ 2 .

Thus, we deduce that

( Φ ′ ( z 1 ) − Φ ′ ( z 2 ) , z 1 − z 2 ) ≥ min { 1 2 p + , 1 2 q + } ( min { ‖ u 1 − u 2 ‖ e p p + , ‖ u 1 − u 2 ‖ e p p − }     + min { ‖ v 1 − v 2 ‖ e q q + , ‖ v 1 − v 2 ‖ e q q − } ) ,

for any z₁ = (u₁, v₁), z₂ = (u₂, v₂) ∈ X, i.e.,Φ' is uniformly monotone.

From (3), (4), we can see that for any z ∈ X, we have that

( Φ ′ ( z ) , z ) ‖ z ‖ ≥ min { ‖ u ‖ e p p + , | u | | e p p − } + min { ‖ v ‖ e p p + , ‖ v ‖ e p p − } ‖ u ‖ e p + ‖ v ‖ e p .

That's meaning Φ' is coercive on X.

By a standard argument, we know that Φ' is hemicontinuous. Therefore, the conclusion follows immediately by applying Theorem 26.A 31.

To obtain our main result, we assume the following conditions on F(x,s,t):

(A1) There exist d(x) ∈ L¹(Ω) and 0 < ς < p^-, 0 < τ < q^- such that

F ( x , s , t ) ≤ d ( x ) 1 + s ζ + t τ

for a.e.x ∈ Ω and (s,t) ∈ R × R;

(A2) F(x,0,0) = 0 for a.e.x ∈ Ω;

(A3) There exist s₁,t₁ ∈ R with |s₁|, |t₁| ≥ 1 such that

meas ( Ω ) sup x , s , t ∈ Ω × [ 0 , c k p ] × [ 0 , c k q ] F ( x , s , t ) ≤ e p 1 p + + e q 1 q + ∫ Ω F ( x , s 1 , s 1 ) d x e p 1 p - s 1 p + + e q 1 q - t 1 q + ,

where c is given in (5) and

k p = max e p 1 + p + e q 1 q + 1 p + , e p 1 + p + e q 1 q + 1 p - , k q = max q + e p 1 p + + e q 1 1 q + , q + e p 1 p + + e q 1 1 q - .

(A3)' F(x,s,t) > 0 for any x ∈ Ω and |s| or |t| large enough, and there exist M, N > 0 such that

F ( x , s , t ) ≤ 0 , x ∈ Ω , s ≤ M , t ≤ N ;

Then we have the following main theorem.

Theorem 2. Assume (A1),(A2),(A3)(or (A3)'),(G) and (E) hold. Then there exist an open interval Λ ⊆ [0, ∞) and a positive real number ρ with the following property: for each λ ∈ Λ, there exists σ > 0 such that for each μ ∈ [0, σ], problem (1) has at least three weak solutions whose norms are less than ρ.

Proof. By the definitions of Φ, Ψ, J, we know that Ψ' is compact, Φ is weakly lower semi-continuous and bounded on each bounded subset of X. From lemma 1 we can see that (Φ')^-1 is well defined, from condition (G), J is well defined and continuously Gâteaux differentiable on X, with compact derivative. Then we can use Theorem 1 to obtain the result. Now we show that the hypotheses of Theorem 1 are fulfilled.

Thanks to (A1), for each λ ≥ 0, one has that

lim z → ∞ Φ ( z ) + λ Ψ ( z ) = + ∞ ,

and so the assumption (i) of Theorem 1 holds.

Now we consider in two cases:

Case (i): (A3) holds, i.e., there exist 1 ≤ |s₁|, |t₁| such that (9) hold.

Now we set z₀ = (0,0), z₁ = (s₁, s₁) and denote r = e p 1 p + + e q 1 q + > 0 , then it is easy to see that

Φ ( z 1 ) > r > 0 = Φ ( z 0 ) .

Thus, (ii) of Theorem 1 is satisfied.

At last, by (A2) we know Ψ(z₀) = 0, then

Φ ( z 1 ) - r Ψ ( z 0 ) + ( r - Φ ( z 0 ) ) Ψ ( z 1 ) Φ ( z 1 ) - Φ ( z 0 ) = r Ψ ( z 1 ) Φ ( z 1 ) ≤ - r ∫ Ω F ( x , s 1 , s 1 ) d x s 1 p + p - e p 1 + t 1 q + q - e q 1 .

On the other way, when Φ(z) ≤ r, we have

min { ‖ u ‖ e p p + , ‖ u ‖ e p p − } ≤ r p + , min { ‖ v ‖ e q p + , ‖ v ‖ e q p − } ≤ r q + .

We deduce that

u e p ≤ max ( r p + ) 1 p + , ( r p + ) 1 p -

and

v e q ≤ max ( r q + ) 1 q + , ( r q + ) 1 q -

For r = e p 1 p + + e q 1 q + , then we have

u e p ≤ k p , v e q ≤ k q .

By (5), we obtain

u ∞ ≤ c k p , v ∞ ≤ c k q .

Thus, from (7), we have

- inf z ∈ Φ - 1 ( - ∞ , r ) Ψ ( z ) = sup z ∈ Φ - 1 ( - ∞ , r ) - Ψ ( z ) ≤ ∫ Ω sup u , v ∈ [ 0 , c k p ] × [ 0 , c k q ] F ( x , u , v ) d x ≤ meas ( Ω ) sup x , u , v ∈ Ω × [ 0 , c k p ] × [ 0 , c k q ] F ( x , u , v )

From (9)-(11) and the definition of r, we can see (iii) of Theorem 1 is hold.

Case (ii): (A3)' holds. Then there exist |s₂|,|t₂| > 1 such that F(x,s₂,t₂) > 0 for any x ∈ Ω and s 2 p - e p 1 ≥ 1 , t 2 q - e q 1 ≥ 1 . Set a = min{c,M}, b = min{c, N} then we have

∫ Ω sup s , t ∈ [ 0 , a ] × [ 0 , b ] F ( x , s , t ) d x ≤ 0 < ∫ Ω F ( x , s 2 , t 2 ) d x .

We denote z₂ = (s₂,t₂) and r = min 1 p + a c p + , 1 q + b c q + . Then it is easy to see that

Φ ( z 2 ) > r > Φ ( z 0 ) .

So, (ii) of Theorem 1 is satisfied.

When Φ(z) ≤ r, similar to the above arguments, we obtain that

u ∞ ≤ a , v ∞ ≤ b .

At last, we see that

Φ ( z 2 ) - r Ψ ( z 0 ) + ( r - Φ ( z 0 ) ) Ψ ( z 2 ) Φ ( z 2 ) - Φ ( z 0 ) = r Ψ ( z 2 ) Φ ( z 2 ) ≤ - r ∫ Ω F ( x , s 2 , t 2 ) d x s 2 p + p - e p 1 + t 2 q + q - e q 1 < 0 .

From (7) and (12), we have

- inf z ∈ Φ - 1 ( - ∞ , r ) Ψ ( z ) = sup z ∈ Φ - 1 ( - ∞ , r ) - Ψ ( z ) ≤ ∫ Ω sup u , v ∈ [ 0 , a ] × [ 0 , b ] F ( x , u , v ) d x ≤ 0 .

From (14) and (15), we can see (iii) of Theorem 1 is still hold.

Then all the hypotheses of Theorem 1 are fulfilled. By Theorem 1, we know that there exist an open interval Λ ⊆ [0, ∞) and a positive constant ρ such that for any λ ∈ Λ, there exists σ > 0 and for each μ ∈ [0,σ], problem (1) has at least three weak solutions whose norms are less than ρ.

By Theorem 2, we have the following result.

Corollary 1. Let f, g : Ω × R → R be Carathéodory functions, sup_|ζ|≤s |g(·, ζ)| ∈ L¹(Ω) for all s > 0, and define F ( x , t ) = ∫ 0 t f ( x , y ) d y for any (x,t) ∈ Ω × R, e(x) ∈ L^∞(Ω) and ess inf_Ωe(x) > 0. Assume the following conditions hold.

(B1) There exist d(x) ∈ L¹(Ω) and 0 < ς < p^- such that

F ( x , t ) ≤ d ( x ) 1 + t ς

for a.e.x ∈ Ω and t ∈ R;

(B2) There exists t₃ ∈ R with |t₃| ≥ 1 such that

m e a s ( Ω ) sup x , s ∈ Ω × [ 0 , c k ] F ( x , s ) ≤ p - p + ∫ Ω F ( x , t 3 ) d x t 3 p + ,

where c is given in (5) and

k = max e 1 1 p + , e 1 1 p - ;

or

(B2)' F(x,t) > 0 for any x ∈ Ω and |t| large enough, and there exist M > 0 such that

F ( x , t ) ≤ 0 , x ∈ Ω , t ≤ M .

Then there exist an open interval Λ ⊆ [0, ∞) and a positive constant ρ such that for any λ ∈ Λ, there exists σ > 0 and for each μ ∈ [0, σ], the problem

- Δ p ( x ) u + e ( x ) u p ( x ) - 2 u = λ f ( x , u ) + μ g ( x , u ) , x ∈ Ω , u = 0 x ∈ ∂ Ω ,

has at least three weak solutions whose norms are less than ρ.

Remark 1. if p(x) = p in Ω, μ = 0, problem (17) was considered in 21. If we take f(x,t) = |t|^γ(x)-2t - t with γ ( x ) ∈ C 0 ( Ω ̄ ) satisfies 2 < γ^- ≤ γ⁺ < p^-, μ = 0, Corollary 1 becomes a version of Theorem 2 in 23. Hence our Corollary 1 unifies and generalizes Theorem 2 in 21 and Theorem 2 in 23 and our Theorem 2 generalizes the main results of 2122232425 to the system (1).

At last, we give two examples.

Example 1. Let Ω = B(0,1) be the unit ball of R^N with N ≥ 2, set p(x) = N + e^|x|,q(x) = N + 1 + ln(1 + x²), e_p(x) = (1 + x²) = e_q(x), G(x,u,v) = x²(u² + v²) and

F ( x , u , v ) = e x 2 ( e u + u v - 1 ) , x ∈ Ω , u ≤ M , v ∈ R , e x 2 u e M + u v + 1 2 u 2 - M u - ( M - 1 ) e M + 1 2 M 2 , x ∈ Ω , u ≤ M , v ∈ R ,

where M is a positive constant, i.e., we consider the following problem

- Δ p ( x ) u + ( 1 + x 2 ) u p ( x ) - 2 u = λ f ( x , u , v ) + μ 2 x 2 u , x ∈ Ω , - Δ q ( x ) v + ( 1 + x 2 ) v q ( x ) - 2 v = λ u + μ 2 x 2 v , x ∈ Ω , ∂ u ∂ v = ∂ v ∂ v = 0 x ∈ ∂ Ω .

where

f ( x , u , v ) = F u ( x , u , v ) = e x 2 ( e u + v ) , x ∈ Ω , u ≤ M , v ∈ R , e x 2 ( e M + v + u - M ) , x ∈ Ω , u ≤ M , v ∈ R ,

We can see that p + = N + e , p - = N + 1 , q + = N + 1 + ln 2 , q - = N + 1 , e 1 = 4 3 , and it is easy to see that for any t₁ > 1, there exists s₁ > 1 such that

e s 1 + s 1 t 1 - 1 s 1 p + p - + t 1 q + q - ≥ e e c k p + c 2 k p k q - 1 1 p + + 1 q + ,

were k p = 4 3 + 4 ( N + 1 + ln + 2 ) 3 ( N + e ) 1 N + 1 , k q = 4 3 + 4 ( N + e ) 3 ( N + 1 + ln 2 ) 1 N + 1 are positive constants and c is given by (5). Then when M ≥ s₁, F(x,u,v) defined in (18) satisfies (A1)-(A3) of Theorem 2, and G(x,u,v),e(x) satisfy

(G) and (E) respectively, by Theorem 2, there exist an open interval Λ ⊆ [0, ∞) and a positive constant ρ such that for any λ ∈ Λ, there exists σ > 0 and for each μ ∈ [0,σ], system (19) has at least three weak solutions whose norms are less than ρ.

Example 2. Assume Ω,p(x),q(x),e_p(x),e_q(x),G(x,u,v) are the same as in example 1, and suppose N ≥ 8. Let

F ( x , u , v ) = ( 1 + 2 x 2 ) u 4 v 2 + v 4 u 2 - 2 u 2 v 2 , x ∈ Ω , u , v ∈ R .

Obviously, F(x,u,v) satisfies (A1) and (A2). By simple computation, we can see that

F ( x , u , v ) > 0 , when u > 2 or v > 2

and

F ( x , u , v ) < 0 , when u < 1 and v < 1 ,

i.e., (A3)' hold for F(x,u,v) defined in (22).

Thus, there exist an open interval Λ ⊆ [0, ∞) and a positive constant ρ such that for any λ ∈ Λ, there exists σ > 0 and for each μ ∈ [0, σ], the system

- Δ p ( x ) u + ( 1 + x 2 ) u p ( x ) - 2 u = λ 4 u 3 v 2 + 2 v 4 u - 4 u v 2 + μ 2 x 2 u , x ∈ Ω , - Δ p ( x ) v + ( 1 + x 2 ) v q ( x ) - 2 v = λ 4 u 3 v 2 + 2 u 4 v - 4 v u 2 + μ 2 x 2 v , x ∈ Ω , ∂ u ∂ v = ∂ v ∂ v = 0 x ∈ ∂ Ω .

has at least three weak solutions whose norms are less than ρ.

Remark 2. We remark that the methods used in this paper are also applicable for the cases of the other boundary value conditions, for example, Dirichlet boundary value conditions.

The authors declare that they have no competing interests.

All authors read and approved the final manuscript.