The exact boundary blow-up rate of large solutions for semilinear elliptic problems

Abstract

In this paper, we establish the blow-up rate of the large positive solution of the singular boundary value problem − △ u = λ u − a ( x ) u p , u | ∂ Ω = + ∞ , where Ω is a bounded smooth domain in R N . The weight function a ( x ) in front of the nonlinearity can vanish on the boundary of the domain Ω at different rates according to the point x 0 of the boundary. The decay rate of the weight function a ( x ) may not be approximated by a power function of distance near the boundary ∂ Ω . We combine the localization method of [J. López-Gómez, The boundary blow-up rate of large solutions, J. Differential Equations 195 (2003) 25–45] with some previous radially symmetric results of [T. Ouyang, Z. Xie, The uniqueness of blow-up solution for radially symmetric semilinear elliptic equation, Nonlinear Anal. 64 (9) (2006) 2129–2142] to prove that any large solution u ( x ) must satisfy lim x → x 0 u ( x ) K ( b x 0 ∗ ( dist ( x , ∂ Ω ) ) ) − β = 1 for each x 0 ∈ ∂ Ω , where b x 0 ∗ ( r ) = ∫ 0 r ∫ 0 s b x 0 ( t ) d t d s , K = [ β ( ( β + 1 ) C 0 − 1 ) ] 1 p − 1 , β = 1 p − 1 , C 0 = lim r → 0 ( ∫ 0 r b x 0 ( t ) d t ) 2 b x 0 ∗ ( r ) b x 0 ( r ) and b x 0 ( r ) is the boundary normal section of a ( x ) at x 0 ∈ ∂ Ω , i.e., b x 0 ( r ) = a ( x 0 − r n x 0 ) , r > 0 , r ∼ 0 , and n x 0 stands for the outward unit normal vector at x 0 ∈ ∂ Ω .