Bubbling along boundary geodesics near the second critical exponent (Q604444)

Let \(\Omega\) be a bounded domain with smooth boundary in \({\mathbb R}^n\) and \(p>1\). This paper deals with the study of positive solutions to the Lane-Emden-Fowler (LEF) problem \(\Delta u+u^p=0\) in \(\Omega\), subject to the Dirichlet boundary condition \(u=0\) on \(\partial\Omega\). The purpose of this paper is to construct solutions of problem (LEF) when \(p\) is below but sufficiently close to the (supercritical) second critical exponent \((n+1)/(n-3)\). Assuming that \(\partial\Omega\) contains a non-degenerate, closed geodesic \(\Gamma\) with strictly negative curvature, the authors find a solution of problem (LEF) with a concentration behavior as \(p\) approaches \((n+1)/(n-3)\) in the form of a bubbling line, eventually collapsing onto \(\Gamma\). This geometric condition holds if for instance \(\Omega\) has a convex hole or it is a deformations of a torus-like solid of revolution like Passaseo's domain. The proofs combine geometric arguments with refined techniques in asymptotic analysis.