In this article we study the existence of nontrivial solutions for the fourth-order boundary value problem

Δ 2 u - M ∫ Ω | ∇ u | 2 d x Δ u = f ( x , u ) , in Ω , u = Δ u = 0 , on ∂ Ω ,

where Ω ⊂ R^N is a bounded smooth domain, f : Ω × R → R and M : R → R are continuous functions. The existence and multiplicity results for Equation (1) are considered in 123 by using variational methods and fixed point theorems in cones of ordered Banach space with space dimension is one.

On the other hand, The four-order semilinear elliptic problem

Δ 2 u + c Δ u = f ( x , u ) , in Ω , u = Δ u = 0 , on ∂ Ω ,

arises in the study of traveling waves in a suspension bridge, or the study of the static deflection of an elastic plate in a fluid, and has been studied by many authors, see 45678910 and the references therein.

Inspired by the above references, the object of this article is to study existence and multiplicity of nontrivial solution of a fourth-order elliptic equation under some conditions on the function M(t) and the nonlinearity. The proof is based on the mountain pass theorem, namely,

Lemma 1.1. Let E be a real Banach space, and I ∈ C¹(E, R) satisfy (PS)-condition. Suppose

(1) There exist ρ > 0, α > 0 such that

I | ∂ B ρ ≥ I ( 0 ) + α

where B_p = {u ∈ E|∥u∥ ≤ ρ}.

(2) There is an e ∈ E and ∥e∥ > ρ such that

I ( e ) ≤ I ( 0 ) .

Then I(u) has a critical value c which can be characterized as

C = inf γ ∈ Γ max u ∈ γ ( [ 0 , 1 ] ) I ( u ) ,

where Γ = {γ ∈ C([0, 1],E)|γ(0) = 0,γ(1) = e}.

The article is organized as follows: Section 2 is devoted to giving the main result and proving the existence of nontrivial solution of Equation (1). In Section 3, we deal with the multiplicity results of Equation (1) whose nonlinear term is asymptotically linear at both zero and infinity

Theorem 2.1. Assume the function M(t) and the nonlinearity f(x, t) satisfying the following conditions:

(H1) M(t) is continuous and satisfies

M ( t ) > m 0 , ∀ t > 0 ,

for some m₀ > 0. In addition, that there exist m' > m₀ and t₀ > 0, such that

M ( t ) = m ′ , ∀ t > t 0 .

(H2) f(x, t) ∈ C(Ω × R); f(x, t) ≡ 0, ∀x ∈ Ω, t ≤ 0, f(x, t) ≥ 0, ∀x ∈ Ω, t > 0;

(H3) |f(x, t)| ≤ a(x) + b|t|^p, ∀t ∈ R and a.e. x in Ω, where a(x) ∈ L^q (Ω), b ∈ R

and 1 < p < N + 4 N - 4 if N > 4 and 1 < p < ∞ if N ≤ 4 and 1 q + 1 p = 1 ;

(H4) f(x, t) = o(|t|) as t → 0 uniformly for x ∈ Ω ;

(H5) There exists a constant Θ > 2 and R > 0, such that

Θ F ( x , s ) ≤ s f ( x , s ) , ∀ | s | ≥ R .

Then Equation (1) has at least one nonnegative solution.

Let Ω ⊂ R^N be a bounded smooth open domain, H = H 2 ( Ω ) ⋂ H 0 1 ( Ω ) be the Hilbert space equipped with the inner product

( u , v ) - ∫ Ω ( Δ u Δ v + ∇ u ∇ v ) d x ,

and the deduced norm

| | u | | 2 = ∫ Ω | Δ u | 2 d x + ∫ Ω | ∇ u | 2 d x .

Let λ₁ be the positive first eigenvalue of the following second eigenvalue problem

- Δ v = λ v , in Ω , v = 0 , on ∂ Ω .

Then from 4, it is clear to see that Λ₁ = λ₁(λ₁ - c) is the positive first eigenvalue of the following fourth-order eigenvalue problem

Δ 2 u + c Δ u = λ u , in Ω , Δ u = u = 0 , on ∂ Ω ,

where c < λ₁. By Poincare inequality, for all u ∈ H, we have

| | u | | 2 ≥ Λ 1 | | u | | L 2 2 .

A function u ∈ H is called a weak solution of Equation (1) if

∫ Ω Δ u Δ v d x + M ∫ Ω | ∇ u | 2 d x ∫ Ω ∇ u ∇ v d x = ∫ Ω f ( x , u ) v d x

holds for any v ∈ H. In addition, we see that weak solutions of Equation (1) are critical points of the functional I : H → R defined by

I ( u ) = 1 2 ∫ Ω | Δ u | 2 d x + 1 2 M ^ ∫ Ω | ∇ u | 2 d x - ∫ Ω F ( x , u ) d x ,

where M ^ ( t ) = ∫ 0 t M ( s ) d s and F(x, t) = ∫ f(x, t)dt. Since M is continuous and f has subcritical growth, the above functional is of class C¹ in H. We shall apply the famous mountain pass theorem to show the existence of a nontrivial critical point of functional I(u).

Lemma 2.2. Assume that (H1)-(H5) hold, then I(u) satisfies the (PS)-condition.

Proof. Let {u_n} ⊂ H be a (PS)-sequence. In particular, {u_n} satisfies

I ( u n ) → C , and I ′ ( u n ) , u n → 0 as n → ∞ .

Since f(x, t) is sub-critical by (H3), from the compactness of Sobolev embedding and, following the standard processes we know that to show that I verifies (PS)-condition it is enough to prove that {u_n} is bounded in H. By contradiction, assume that ∥u_n∥ → +∞.

Case I. If ∫ Ω | ∇ u n | 2 d x is bounded, ∫ Ω | Δ u n | 2 d x → + ∞ . We assume that there exist a constant K > 0 such that ∫ Ω | ∇ u n | 2 d x ≤ K . By (H1), it is easy to obtain that m ̃ = max t ∈ [ 0 , K ] M ( t ) > m 0 . Set l 1 = min { 1 , m 0 } , l 2 = max { 1 , m ̃ } . Then, from

(H1), (H3), and (H5), we have

I ( u n ) - l 1 2 l 2 I ′ ( u n ) u n = 1 2 ∫ Ω | Δ u n | 2 d x + 1 2 M ^ ∫ Ω | ∇ u n | 2 d x - ∫ Ω F ( x , u n ) d x - l 1 2 l 2 ∫ Ω | Δ u n | 2 d x + M ∫ Ω | ∇ u n | 2 d x ∫ Ω | ∇ u n | 2 d x + l 1 2 l 2 ∫ Ω f ( x , u n ) u n d x ≥ 1 2 l 1 | | u n | | 2 + ∫ Ω l 1 2 l 2 f ( x , u n + ) u n - F ( x , u n + ) d x ≥ 1 2 l 1 | | u n | | 2 + ∫ | | u n | | ≥ R l 1 2 l 2 f ( x , u n + ) u n + - F ( x , u n + ) d x - C 1 ≥ 1 2 l 1 | | u n | | 2 + l 1 2 l 2 ∫ | | u n | | ≥ R f ( x , u n + ) u n + - 2 l 2 l 2 F ( x , u n + ) d x - C 1 ≥ 1 2 l 1 | | u n | | 2 + l 1 2 l 2 ∫ | | u n | | ≥ R f ( x , u n + ) u n + - Θ F ( x , u n + ) d x - C 1 .

On the other hand, it is easy to obtain that

I ( u n ) - l 1 2 l 2 I ′ ( u n ) u n ≤ C + C | | u n | | .

Then, from above, we can have

| | u | | 2 ≤ C + C | | u n | | ,

which contradicts ∥u_n∥ → +∞. Therefore {u_n} is bounded in H.

Case II. if ∫ Ω|Δun|2dx → +∞ . By (H1), let l₂ = max{1, m'}, we also can obtain that {u_n} is bounded in H.

This lemma is completely proved.

Lemma 2.3. Suppose that (H1)-(H5) hold, then we have

(1) there exist constants ρ > 0, α > 0 such that I | ∂ B ρ ≥ α with B_p = {u ∈ H ∥u∥ ≤ ρ};

(2) I(tφ₁) → -∞ as t → +∞.

Proof. By (H1)-(H4), we see that for any ε > 0, there exist constants C ₁ > 0, C₂ such that for all (x, s) ∈ Ω × R, one have

F ( x , s ) ≤ 1 2 ε s 2 + C 1 s p + 1

Choosing ε > 0 small enough, we have

I ( u ) = 1 2 ∫ Ω | Δ u | 2 d x + 1 2 M ^ ∫ Ω | ∇ u | 2 d x - ∫ Ω F ( x , u ) d x ≥ 1 2 ∫ Ω | Δ u | 2 d x + 1 2 m 0 ∫ Ω | ∇ u | 2 d x - ∫ Ω F ( x , u ) d x ≥ 1 2 l 1 | | u | | 2 - ε 2 | | u | | L 2 2 - C 1 | | u | | L p + 1 p + 1 ≥ 1 2 ( l 1 - ε ) | | u | | 2 - C 3 | | u | | p + 1 .

by (3), (5), (7) and the Sobolev inequality. So, part 1 is proved if we choose ∥u∥ ρ > 0 small enough.

On the other hand, we have

I ( u ) = 1 2 ∫ Ω | Δ u | 2 d x + 1 2 M ^ ∫ Ω | ∇ u | 2 d x - ∫ Ω F ( x , u ) d x ≤ 1 2 ∫ Ω | Δ u | 2 d x + 1 2 m 1 ∫ Ω | ∇ u | 2 d x - ∫ Ω F ( x , u ) d x ≥ 1 2 l 2 | | u | | 2 - | | u | | Θ Θ + C 4 .

using (4) and (H5). Hence,

I ( t φ 1 ) ≤ 1 2 l 2 t 2 | | φ 1 | | 2 - t Θ | | φ 1 | | Θ Θ + C 4 → - ∞

as t → +∞ and part 2 is proved.

Proof of Theorem 2.1. From Lemmas 2.2 and 2.3, it is clear to see that I(u) satisfies the hypotheses of Lemma 1.1. Therefore I(u) has a critical point.

Theorem 3.1. Assume that (H1) holds. In addition, assume the following conditions are hold:

(H6) f(x, t)t ≥ 0 for x ∈ Ω, t ∈ R;

(H7) lim t → 0 f ( x , t ) t = α , lim | t | → + ∞ f ( x , t ) t = β , uniformly in a.e x ∈ Ω, where α min { 1 , m 0 } < λ 1 ( λ 1 + m ′ ) < β < + ∞ .

Then Equation (1) has at least two nontrivial solutions, one of which is positive and the other is negative.

Let u⁺ = max{u, 0}, u^- = min{u, 0}. Consider the following problem

Δ 2 u - M ∫ Ω | ∇ u | 2 d x Δ u = f + ( x , u ) , in Ω , u = Δ u = 0 , on ∂ Ω ,

where

f + ( x , t ) = f ( x , t ) if t ≥ 0 , 0 , if t < 0 .

Define the corresponding functional I⁺ : H → R as follows:

I + ( u ) = 1 2 ∫ Ω | Δ u | 2 d x + 1 2 M ^ ∫ Ω | ∇ u | 2 d x - ∫ Ω F + ( x , u ) d x , ∀ u ∈ H ,

where F + ( x , u ) = ∫ 0 u f + ( x , t ) d t . Obviously, I⁺ ∈ C^l(H, R). Let u be a critical point of I⁺ which implies that u is the weak solution of Equation (8). Futhermore, by the weak maximum principle it follows that u ≥ 0 in Ω. Thus u is also a solution of Equation (1).

Similarly, we also can define

f - ( x , t ) = f ( x , t ) if t ≤ 0 , 0 , if t < 0 .

and

I - ( u ) = 1 2 ∫ Ω | Δ u | 2 d x + 1 2 M ^ ∫ Ω | ∇ u | 2 d x - ∫ Ω F - ( x , u ) d x , ∀ u ∈ H ,

where F - ( x , u ) = ∫ 0 u f - ( x , t ) d t . Obviously, I^- ∈ C¹(H, R). Let u be a critical point of I^- which implies that u is the weak solution of Equation (1) with I^-(u) = I(u).

Lemma 3.2. Assume that (H1), (H6), and (H7) hold, then I^± satisfies the (PS) condition.

Proof. We just prove the case of I⁺. The arguments for the case of I^- are similar. Since Ω is bounded and (H7) holds, then if {u_n} is bounded in H, by using the Sobolve embedding and the standard procedures, we can get a convergent subsequence. So we need only to show that {u_n} is bounded in H.

Let {u_n} ⊂ H be a sequence such that

I + ( u n ) → c , ∇ I + ( u n ) → 0 .

By (H7), it is easy to see that

| f + ( x , s ) s | ≤ C ( 1 + | s | 2 ) .

Now, (9) implies that, for all ϕ ∈ H, we have

∫ Ω Δ u n Δ ϕ d x + M ∫ Ω | ∇ u n | 2 d x ∫ Ω ∇ u n ∇ ϕ d x = ∫ Ω f + ( x , u n ) ϕ d x → 0 .

Set ϕ = u_n, we have

min { 1 , m 0 } | | u n | | 2 ≤ ∫ Ω | Δ u n | 2 d x + M ∫ Ω | ∇ u n | 2 d x ∫ Ω | ∇ u n | 2 d x = ∫ Ω f + ( x , u n ) u n d x + ∇ I + ( u n ) , u n ≤ ∫ Ω f + ( x , u n ) u n d x + o ( 1 ) | | u n | | ≤ C + C | | u n | | L 2 2 + o ( 1 ) | | u n | | .

Next, we will show that | | u n | | L 2 2 is bounded. If not, we may assume that ∥u_n∥_L2 → +∞ as n → +∞. Let ω n = u n | | u n | | L 2 , then | | ω n | | L 2 = 1 . From (11), we have

| | ω n | | 2 ≤ o ( 1 ) + C + o ( 1 ) | | u n | | L 2 | | u n | | | | u n | | L 2 = o ( 1 ) + C + o ( 1 ) | | ω n | | ,

thus {ω_n} is bounded in H. Passing to a subsequence, we may assume that there exists ω ∈ H with | | ω | | L 2 = 1 such that

ω n ⇀ ω , weakly in H , n → + ∞ , ω n → ω , strongly in L 2 ( Ω ) , n → + ∞ .

On the other hand, | | u n | | L 2 → + ∞ as n → +∞, by Poincare inequality, it is easy to know that ∫ Ω|Δun|2dx → +∞ as n → +∞. Thus by (H1), the function M ∫ Ω | ∇ u n | 2 d x = m ′ . So as n → +∞, by (10), we have

∫ Ω Δ ω Δ ϕ d x + m ′ ∫ Ω ∇ ω ∇ ϕ d x ∫ Ω β ω + ϕ d x = 0 , ∀ ϕ ∈ H .

Then ω ∈ H is a weak solution of the equation

Δ 2 ω - m ′ Δ ω = β ω + .

The weak maximum principle implies that ω = ω⁺ ≥ 0. Choosing ϕ (x) = φ₁(x) > 0, which is the corresponding eigenfunctions of λ₁. From (10), we get

∫ Ω Δ ω Δ φ 1 d x + m ′ ∫ Ω ∇ ω ∇ φ 1 d x = β ∫ Ω ω + φ 1 d x .

On the other hand, we can easily see that Λ = λ₁(λ₁ + m') is the eigenvalue of the problem

Δ 2 u + m ′ Δ u = Λ u , in Ω , Δ u = u = 0 , on ∂ Ω

and the corresponding eigenfunction is still φ₁(x). If ω(x) > 0, we also have

∫ Ω Δ ω Δ φ 1 d x + m ′ ∫ Ω ∇ ω ∇ φ 1 d x = Λ ∫ Ω ω + φ 1 d x ,

which follows that ω ≡ 0 by Λ < β But this conclusion contradicts ||ω||L2 = 1 .

Hence {u_n} is bounded in H.

Now we prove that the functionals I^± has a mountain pass geometry.

Lemma 3.3. Assume that (H1), (H7) hold, then we have

(1) there exists ρ, R > 0 such that I^±(u) > R, if ∥u∥ = ρ;

(2) I^±(u) are unbounded from below.

Proof. By (H7), for any ε > 0, there exists C ₁ > 0, C₂ > 0 such that ∀(x, s) ∈ Ω × R, we have

F ( x , s ) ≤ 1 2 ( α + ε ) s 2 + C 1 s p + 1

and

F ( x , s ) ≥ 1 2 ( β + ε ) s 2 + C 2 ,

where 2 < p < 2 * = 2 N N - 2 N > 2 , + ∞ N ≤ 2 .

We just prove the case of I⁺. The arguments for the case of I^- are similar. Let ϕ = tφ₁. When t is sufficiently large, by (16) and (H1), it is easy to see that

I + ( t φ 1 ) = 1 2 ∫ Ω | Δ ( t φ 1 ) | 2 d x + 1 2 M ^ ∫ Ω | ∇ ( t φ 1 ) | 2 d x - ∫ Ω F + ( x , t φ 1 ) d x ≤ 1 2 ∫ Ω | Δ ( t φ 1 ) | 2 d x + 1 2 m ′ ∫ Ω | ∇ ( t φ 1 ) | 2 d x - ∫ Ω 1 2 ( β - ε ) ( t φ 1 ) 2 - C 2 d x = t 2 2 ∫ Ω | Δ φ 1 | 2 d x + m ′ ∫ Ω | Δ φ 1 | 2 d x - ( β - ε ) ∫ Ω φ 1 2 d x + C 2 | Ω | = t 2 2 [ Λ - ( β - ε ) ] | | φ 1 | | L 2 + C 2 | Ω | → - ∞ , a s t → + ∞ .

On the other hand, by (17), (H1), the Poincare inequality and the Sobolve embedding, we have

I + ( u ) = 1 2 ∫ Ω | Δ u | 2 d x + 1 2 M ^ ∫ Ω | ∇ u | 2 d x - ∫ Ω F + ( x , u ) d x ≥ 1 2 min { 1 , m 0 } | | u | | - α + ε 2 ∫ Ω | u | 2 d x - C 1 ∫ Ω | u | p + 1 d x ≥ 1 2 min { 1 , m 0 } - α + ε 2 Λ | | u | | - C 4 | | u | | p + 1 ,

where C₄ is a constant. Choosing ∥u∥ = ρ small enough, we can obtain I⁺(u) ≥ R > 0 if ∥u∥ = ρ.

Proof of Theorem 3.1. From Lemma 3.3, it is easy to see that there exists e ∈ H with ∥e∥ > ρ such that I^±(e) < 0.

Define

P = { γ : [ 0 , 1 ] → H : γ is continuous and γ ( 0 ) = 0 , γ ( 1 ) = e } ,

and

c ± = inf γ ∈ P max t ∈ [ 0 , 1 ] I ± ( γ ( t ) ) .

From Lemma 3.3, we have

I ± ( 0 ) = 0 , I ± ( e ) < 0 , I ± ( u ) | ∂ B ρ ≥ R > 0 .

Moreover, by Lemma 3.2, the functions I^± satisfies the (PS)-condition. By Lemma 1.1, we know that c⁺ is a critical value of I⁺ and there is at least one nontrivial critical point in H corresponding to this value. This critical in nonnegative, then the strong maximum principle implies that is a positive solution of Equation (1). By an analogous way we know there exists at least one negative solution, which is a nontrivial critical point of I^- Hence, Equation (1) admits at least a positive solution and a negative solution.

The authors declare that they have no competing interests.

In this manuscript the authors studied the existence and multiplicity of solutions for an interesting fourth-order elliptic equation by using the famous mountain pass lemma. Moreover, in this work, the authors' supplements done in 123. All authors typed, read and approved the final manuscript.