Existence and Multiplicity of Nontrivial Solutions for a Class of Fourth-Order Elliptic Equations

Using the Fountain theorem and a version of the Local Linking theorem, we obtain some existence and multiplicity results for a class of fourth-order elliptic equations.

Let  :  → R be the functional defined by where (, ) = ∫ for any , V ∈ , so that a critical point of the functional  in  corresponds to a weak solution of problem (1).
In recent years, fourth-order problems have been studied by many authors.In [1], Lazer and McKenna have pointed out that problem (1) furnishes a model to study travelling waves in suspension bridges if (, ) = (( + 1) + − 1), where  + = max{, 0} and  ∈ R. Since then, more general nonlinear fourth-order elliptic boundary value problems have been studied.
In [2,3], Micheletti and Pistoia proved that problem Δ 2  + Δ =  (, ) , in Ω,  = Δ = 0, on Ω, (8) admits two or three solutions by variational method.In [4], Zhang obtained the existence of weak solutions for problem (8) when (, ) is sublinear at ∞.In [5], Zhang and Li showed that problem (8) has at least two nontrivial solutions by means of Morse theory and local linking.When (, ) is asymptotically linear at infinity, the existence of three nontrivial solutions has been obtained in [6] by using Morse theory.In [7], by using the mountain pass theorem, An and Liu gave the existence result for nontrivial solutions for a class of asymptotically linear fourth-order elliptic equations.
In [8], Zhou and Wu got the existence of four sign-changing solutions or infinitely many sign-changing solutions for (8) by using the sign-changing critical point theorems.In [9], Yang and Zhang showed new results on invariant sets of the gradient flows of the corresponding variational functionals and proved the existence of positive, negative, and signchanging solutions for some fourth-order semilinear elliptic boundary value problems.In [10], by using the variational method, Liu and Huang obtained an existence result of signchanging solutions as well as positive and negative solutions for a fourth-order elliptic problem whose nonlinear term is asymptotically linear at both zero and infinity.
In this paper, we will study the existence of nontrivial solutions of problem (1).Our main results are the following theorems.

Proof of Main Results
In this paper, we will use the Fountain Theorem of Bartsch ([13, Theorem 2.5], [14,Theorem 3.6]) to prove our Theorems 1 and 2. And, we will prove Theorems 4 and 5 by using a version of Local Linking theorem [12, Theorem 2.2] which extends theorems given by Li and Willem [15], Li and Szulkin [16].
In [13,14], Bartsch established the Fountain Theorem under the (PS)  condition.Since the Deformation Theorem is still valid under the Cerami condition, the Fountain Theorem is true under the Cerami condition.So, we have the following Fountain Theorem.
Let  be a reflexive and separable Banach space.It is well known that there exist where  , = 1 for  =  and  , = 0 for  ̸ = . ( Let   = RV  ; then  = ⨁ ≥1   .We define ( 2 ) then  has an unbounded sequence of critical values.
For the reader's convenience, we state the following Local Linking theorem [12,Theorem 2.2].Let  be a real Banach space with  =  1 ⊕  2 and We say that  ∈  1 (, R) satisfies the ( * ) condition if every sequence {   } such that {  } is admissible and satisfies contains a subsequence which converges to a critical point of , where   = |   .
Proof of Theorem 2. Firstly, we claim that  satisfies the Cerami condition (C).Consider a sequence {  } such that (  ) is bounded from above and ‖  (  )‖(1 + ‖  ‖) → 0 as  → ∞.By a standard argument, we only need to prove that {  } is a bounded sequence in .For otherwise, we can assume that ‖  ‖ → ∞ as  → ∞.
Proof of Theorem 4. The proof of this theorem is divided in several steps.
Step 1.We claim that  has a local linking at zero with respect to ( 1 ,  2 ).
By ( 5 ), for any  > 0, there exists  1 > 0 such that We obtain from the above expression and (49) that where  2 =  6 + for all  ∈ .
Here, we consider only the case where 0 is an eigenvalue of Δ 2 +Δ+ and case (ii) of ( 6 ) holds.The case (i) is similar.
Step 2. In a way similar to the proof of Theorem 1, we can get that  satisfies the ( * ) condition.
Step 3. Now, we claim that for each  ∈ N, one has Since dim( 0 ) < ∞ and dim( 1  ) < ∞, all the norms are equivalent.For  ∈  Hence, all the assumptions of Theorem B are verified.Then, the proof of Theorem 4 is completed.