Asymptotic behavior of the least energy solution of a problem with competing powers

We consider the problem ε2Δu−uq+up=0 in Ω, u>0 in Ω, u=0 on ∂Ω. Here Ω is a smooth bounded domain in RN, if N⩾3 and ε is a small positive parameter. We study the asymptotic behavior of the least energy solution as ε goes to zero in the case . We show that the limiting behavior is dominated by the singular solution ΔG−Gq=0 in Ω\{P}, G=0 on ∂Ω. The reduced energy is of nonlocal type.