Uniqueness and blow-up rate of large solutions for elliptic equation [formula omitted

Abstract

In this paper, we establish the blow-up rate of the large positive solution of the singular boundary value problem { − Δ u = λ u − b ( x ) h ( u ) in Ω , u = + ∞ on ∂ Ω , where Ω is a smooth bounded domain in R N . The weight function b ( x ) is a non-negative continuous function in the domain. h ( u ) is locally Lipschitz continuous and h ( u ) / u is increasing on ( 0 , ∞ ) and h ( u ) ∼ H u p for sufficiently large u with H > 0 and p > 1 . Naturally, the blow-up rate of the problem equals its blow-up rate for the very special, but important, case when h ( u ) = H u p . We distinguish two cases: (I) Ω is a ball domain and b is a radially symmetric function on the domain in Theorem 1.1; (II) Ω is a smooth bounded domain and b satisfies some local condition on each boundary normal section assumed in Theorem 1.2. The blow-up rate is explicitly determined by functions b and h. In case (I), the singular boundary value problem has a unique solution u satisfying lim d ( x ) → 0 u ( x ) K H − β ( b ∗ ( ‖ x − x 0 ‖ ) ) − β = 1 , where d ( x ) = dist ( x , ∂ Ω ) , b ∗ ( r ) and K are defined by b ∗ ( r ) = ∫ r R ∫ s R b ( t ) d t d s , K = [ β ( ( β + 1 ) C 0 − 1 ) ] 1 p − 1 , β : = 1 p − 1 . In case (II), the blow-up rates of the solutions to the boundary value problem are established and the uniqueness is proved.