Sign changing bubbling solutions for a critical Neumann problem in 3D

Abstract

Let Ω be a smooth bounded domain in R3. We consider the problem Δu−λu+|u|4u=0inΩ,with zero Neumann boundary conditions, and investigate the existence of sign changing solutions, which concentrate simultaneously around k different points of Ω as λ approaches a special value λ0. We characterize the number λ0 in terms of the Green function of −Δ+λ. In particular, we construct a family of solutions exhibiting bubbling behavior at two different points of an annular domain as λ tends to λ0. In doing so, we also derive a representation formula of the Green function of −Δ with Neumann boundary conditions in the annulus in terms of zonal harmonics.