On Fourth-Order Elliptic Equations of Kirchhoff Type with Dependence on the Gradient and the Laplacian

We consider a nonlocal fourth-order elliptic equation of Kirchhoff type with dependence on the gradient and Laplacian Δ2u − (a + b ∫ Ω |∇u|2dx)Δu = f(x, u, ∇u, Δu), in Ω, u = 0, Δu = 0, on ∂Ω, where a, b are positive constants. We will show that there exists b∗ > 0 such that the problem has a nontrivial solution for 0 < b < b∗ through an iterative method based on the mountain pass lemma and truncation method developed by De Figueiredo et al., 2004.


Introduction
This paper concerns with the existence of solutions of the fourth-order Kirchhoff type elliptic equation as follows: where Ω is a bounded and smooth domain in   ( ≥ 5), ,  are positive constants, and  : Ω ×  ×   ×  →  is locally Lipschitz continuous.
The fourth-order elliptic equation arises in the study of traveling waves in suspension bridges, which has been extensively investigated in recent years, such as [1][2][3][4][5][6].To our attention, some authors paid more attention to a more general biharmonic elliptic problem Δ 2  + Δ +  ()  =  (, , Δ, ∇) , in Ω,  () = 0, Δ () = 0, on Ω. (3) For this problem, due to the presence of Δ and ∇ in , it is not variational.To overcome this difficulty, in [5], Wang deals with this problem via the upper and lower solutions and monotone iterative methods; in [7], the authors apply a technique developed by De Figueiredo et al. [8,9] in studying a second-order elliptic problem involving the gradient, which "freezes" the gradient, and use truncation on the nonlinearity .Thus the new problem becomes variational and an iterative scheme of the mountain pass "approximated" solutions is built.
In addition, the nonlocal fourth-order equation has also been studied by many authors.We refer the readers to [10][11][12][13][14][15][16][17][18][19][20].Particularly, Wang et al. studied the following fourthorder equation of Kirchhoff type equation where  is a positive parameter.The authors showed that there exists  * such that the fourth-order elliptic equation has a nontrivial solution for 0 <  <  * by using the mountain pass iterative techniques and the truncation method.Motivated by these works, to study problem (1), we combine the famous mountain pass lemma with a technique developed by De Figueiredo et al. [8], which "freezes" the gradient and the Laplacian variable and use truncation on the nonlinearity of .For convenience, we recall a definition and restate the mountain pass theorem.Definition 1.Let  be a real Banach space and  :  →  a  1 -functional.A sequence {  } in  is a (PS)-sequence for  if (  ) →  for some constant  ≥ 0 as  → ∞, while ⟨  (  ),   ⟩ → 0 as  → ∞.We say that the functional  satisfies the (PS)-condition if any (PS)-sequence for  has a convergent subsequence.

The Main Result
where and   ( = 1, 2, 3) are the optimal constants of the following inequalities: where ‖ ⋅ ‖ is the norm of the Hilbert space X = H 2 (Ω) ∩ H 1 0 (Ω) defined by Then there exists  * > 0 such that (1) has at least a nontrivial solution for 0 <  <  * .
Taking an arbitrary V ∈ X with ‖V‖ = 1, then from (23), we get which implies that the second result of Lemma 3 holds.

Lemma 4.
Let R > 0 and  ∈ X be fixed.Then the functional  R  (⋅) satisfies the (PS)-condition.
Proof.Let {  } ⊂ X be a (PS)-sequence; namely, From the standard processes, we only need to prove that {  } is bounded in X.On a contradiction, suppose that ‖  ‖ → +∞; then, from ( 3 ), we obtain On the other hand, from (29) we know that Then, from the above inequalities, we get which contradicts with ‖  ‖ → +∞.Therefore the sequence {  } is bounded in X.
Lemma 5.For any R > 0 and  ∈ X, problem (15) has a nontrivial weak solution.
Proof.By Theorem A, Lemmas 3, and 4, the result holds.
Lemma 6.Let R > 0 be fixed.Then there exist positive constants ] 1 and ] 2 fl ] 2 (R), independent of , such that for every solution  R  obtained in Lemma 5. Proof.

Now let 𝑢 R
( = 1, 2, . ..) be the weak solution of the following problem: with  =  −1 , where  R −1 was found in Lemma 5 and R = R obtained in Lemma 8. From Lemmas 6-8, we have Thus Lemma 9. Assume that ( 4 ) holds.Let Then { R  } strongly converges in X.
Example 10.Consider the following problem: and () and () are positive and continuous functions.It is easy to verify that (, ,  1 ,  2 ) satisfies all the conditions of ( 1 )-( 4 ).

Conclusion
The paper considers a class of fourth-order elliptic equations of Kirchhoff type with dependence on the gradient and Laplacian.The existence of a nontrivial solution of ( 1) is established when we choose appropriate  * such that 0 <  <  * .The paper generalized the conclusions in [7,14] and weakened the condition in [7].In the following research work, we will also consider problem ( 1), but we just truncate the right side of the equation, and the left of the equation remains the same.