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

We consider the problem ε2 u − uq + up = 0 in Ω, u > 0 in Ω, u = 0 on ∂Ω. Here Ω is a smooth bounded domain in RN, 1 < q < p < N+2 N−2 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 q N N−2 . We show that the limiting behavior is dominated by the singular solution G − Gq = 0 in Ω\{P}, G = 0 on ∂Ω. The reduced energy is of nonlocal type. © 2011 Elsevier Inc. All rights reserved