Solutions for perturbed biharmonic equations with critical nonlinearity

In this paper, we study the perturbed biharmonic equationsε 4Δ 2 u + V ( x ) u = P ( x ) | u | p − 2 u + Q ( x ) | u |2 * * − 2 u , x ∈ R N ,u ∈ H 2(R N) , u ( x ) → 0 , as  | x | → ∞ ,where Δ 2 is the biharmonic operator, N ≥ 5 , 2 * * = 2 N N − 4is the Sobolev critical exponent, p  ∈ (2,2  * *  ), P ( x ), and Q ( x ) are bounded positive functions. Under some given conditions on V , we prove that the problem has at least one nontrivial solution provided that ε ≤ E and that for anyn * ∈ N , it has at least n  *  pairs solutions if ε ≤ En *, where E andEn *are sufficiently small positive numbers. Moreover, these solutions u ε  → 0 inH 2(R N) as ε  → 0. Copyright © 2013 The authors. Mathematical Methods in the Applied Sciences published by John Wiley & Sons, Ltd.