On a bi-harmonic equation involving critical exponent: Existence and multiplicity results

In this paper, we consider the problem of existence as well as multiplicity results for a bi-harmonic equation under the Navier boundary conditions: Δ2u = K(x)up, u > 0 in Ω, Δu = u = 0 on ∂Ω, where Ω is a smooth domain in ℝn , n ≥ 5, and is the critical Sobolev exponent. We obtain highlightly a new criterion of existence, which provides existence results for a dense subset of positive functions, and generalizes Bahri-Coron type criterion in dimension six. Our argument gives also estimates on the Morse index of the obtained solutions and extends some known results. Moreover, it provides, for generic K, Morse inequalities at infinity, which delivers lower bounds for the number of solutions. As further applications of this Morse theoretical approach, we prove more existence results.