Sharp Blow-Up Profiles of Positive Solutions for a Class of Semilinear Elliptic Problems

This paper analyzes the behavior of the positive solution theta(epsilon) of the perturbed problem {-Delta u = lambda m(x)u - [a(epsilon)(x) + b(epsilon)(x)]u(p) = 0 in Omega, Bu = 0 on partial derivative Omega, as epsilon down arrow 0, where a(epsilon)(x) approximate to epsilon(alpha) a(x) and b(epsilon)(x) approximate to epsilon(beta)b(x) for some alpha >= 0 and beta >= 0, and some Holder continuous functions a(x) and b(x) such that a >= 0 (i.e., a >= 0 and a a not equivalent to 0) and min(Omega) b > 0. The most intriguing and interesting case arises when a(x) degenerates, in the sense that Omega(0) int a(-1) (0) is a non-empty smooth open subdomain of Omega, as in this case a "blow-up" phenomenon appears due to the spatial degeneracy of a(x) for sufficiently large lambda. In all these cases, the asymptotic behavior of theta(epsilon) will be characterized according to the several admissible values of the parameters alpha and beta. Our study reveals that there may exist two different blow-up speeds for theta(epsilon) in the degenerate case.