Existence and nonexistence of nontrivial solutions for critical biharmonic equations

Abstract In this paper, we consider the existence and nonexistence of nontrivial solutions for the following biharmonic problem with critical exponent { Δ 2 u = μ Δ u + λ u + | u | 2 ⁎ ⁎ − 2 u , x ∈ Ω , u | ∂ Ω = ∂ u ∂ n | ∂ Ω = 0 , where Ω ⊂ R N is a bounded domain with smooth boundary ∂Ω, Δ 2 = Δ Δ denotes the iterated N-dimensional Laplacian, 2 ⁎ ⁎ = 2 N N − 4 ( N > 4 ) is the critical Sobolev exponent for the embedding H 0 2 ( Ω ) ↪ L 2 ⁎ ⁎ ( Ω ) and H 0 2 ( Ω ) is the closure of C 0 ∞ ( Ω ) under the norm | | Δ u | | L 2 ( Ω ) . Under some assumptions on μ and λ, we prove the existence and nonexistence of nontrivial solutions to the above problem. Different from the case μ = 0 , N = 6 , 7 are not the critical dimensions of nontrivial solutions when μ ∈ ( − β ( Ω ) , 0 ) , where β ( Ω ) : = inf u ∈ H 0 2 ( Ω ) ∖ { 0 } ∫ Ω | Δ u | 2 d x ∫ Ω | ∇ u | 2 d x .