Sharp patterns of positive solutions for some weighted semilinear elliptic problems.

This paper deals with the semilinear elliptic problem - Δ u = λ m (x) u - [ a (x) + ε b (x) ] u p in Ω , B u = 0 on ∂ Ω , where p > 1 , λ > 0 , m , a , b ∈ C (Ω ¯) , with a ⪈ 0 , b ⪈ 0 , Ω is a bounded C 2 domain of R N ( N ≥ 1 ), B is a general classical mixed boundary operator, and ε ≥ 0 . Thus, a(x) and b(x) can vanish on some subdomain of Ω and the weight function m(x) can change sign in Ω . Through this paper we are always considering classical solutions. First, we characterize the existence of positive solutions of this problem in the special case when ε = 0 . Then, we investigate the sharp patterns of the positive solutions when ε ↓ 0 and ε ↑ ∞ . Our study reveals how the existence of sharp profiles is determined by the behavior of b(x). [ABSTRACT FROM AUTHOR]