Interior bubbling solutions for the critical Lin-Ni-Takagi problem in dimension 3

We consider the problem of finding positive solutions of the problem Δu − λu + u ⁵ = 0 in a bounded, smooth domain Ω in ℝ ³ , under zero Neumann boundary conditions. Here λ is a positive number. We analyze the role of Green’s function of −Δ + λ in the presence of solutions exhibiting single bubbling behavior at one point of the domain when λ is regarded as a parameter. As a special case of our results, we find and characterize a positive value λ _* such that if λ − λ _* > 0 is sufficiently small, then this problem is solvable by a solution u _λ which blows-up by bubbling at a certain interior point of Ω as λ ↓ λ _* .