Asymptotic behavior of positive solutions of some elliptic problems

In this paper, we discuss the asymptotic behavior of the positive solutions of the problem −∆u = au − bup, u|∂Ω = 0 as p → 1+ 0 and as p→∞. We show that, for each case, the behavior is determined by a limiting problem. Moreover, the limiting problem is of free boundary nature when p→∞. 1. Introduction an