Nonlocal problems at nearly critical growth

We study the asymptotic behavior of solutions to the nonlocal nonlinear equation (−Δp)su=|u|q−2u(−Δp)su=|u|q−2u in a bounded domain Ω⊂RNΩ⊂RN as qq approaches the critical Sobolev exponent p∗=Np/(N−ps)p∗=Np/(N−ps). We prove that ground state solutions concentrate at a single point x̄∈Ω¯ and analyze the asymptotic behavior for sequences of solutions at higher energy levels. In the semi-linear case p=2p=2, we prove that for smooth domains the concentration point x̄ cannot lie on the boundary, and identify its location in the case of annular domains.