On a fourth order elliptic equation with critical nonlinearity in dimension six (Q2492414)

The authors study the boundary value problem \[ \Delta^2u = K(x)u^p,\qquad u>0\;\text{ in }\Omega,\qquad \Delta u = u = 0\;\text{ on }\partial\Omega\tag{\(*\)} \] (the Navier boundary conditions). Here \(\Omega\subset \mathbb R^6\) is a smooth bounded domain, \(p+1=2n/(n-4)=6\) is the critical Sobolev exponent and \(K>0\) in \(\overline\Omega\). The main result asserts that if \(K\) satisfies certain conditions, including a condition on indices of critical points of \(K\), then (\(*\)) has a solution. In order to show this, the authors prove a Morse lemma at infinity and characterize the critical points at infinity (these notions have been introduced by \textit{A. Bahri} in [Critical points at infinity in some variational problems. Pitman Res. Notes Math. 182, Longman, Harlow (1989; Zbl 0676.58021)]). In the final step of the proof of the main result it is shown that if the only critical points of a functional associated with (\(*\)) are those at infinity, then the index condition must be violated.