Bubbling on boundary submanifolds for the Lin-Ni-Takagi problem at higher critical exponents

Let Ω be a bounded domain in ℝnwith smooth boundary ∂Ω. We consider the equation d2Δu-u + u n-κ+2/n-κ+2 = 0 in Ω, under zero Neumann boundary conditions, where d is a small positive parameter. We assume that there is a κ-dimensional closed, embedded minimal submanifold K of ∂Ω which is nondegenerate, and a certain weighted average of sectional curvatures of ∂Ω is positive along K. Then we prove the existence of a sequence d = dj→ 0 and a positive solution udsuch that d2|∇ud|2⇀ SδKas d → 0 in the sense of measures, where δKstands for the Dirac measure supported on K and S is a positive constant. © European Mathematical Society 2014.