Blow-up rate and uniqueness of singular radial solutions for a class of quasi-linear elliptic equations

Abstract

We establish the uniqueness and the blow-up rate of the large positive solution of the quasi-linear elliptic problem − Δ p u = λ u p − 1 − b ( x ) h ( u ) in B R ( x 0 ) with boundary condition u = + ∞ on ∂ B R ( x 0 ) , where B R ( x 0 ) is a ball centered at x 0 ∈ R N with radius R, N ⩾ 3 , 2 ⩽ p < ∞ , λ > 0 are constants and the weight function b is a positive radially symmetrical function. We only require h ( u ) to be a locally Lipschitz function with h ( u ) / u p − 1 increasing on ( 0 , ∞ ) and h ( u ) ∼ u q − 1 for large u with q > p − 1 . Our results extend the previous work [Z. Xie, Uniqueness and blow-up rate of large solutions for elliptic equation − Δ u = λ u − b ( x ) h ( u ) , J. Differential Equations 247 (2009) 344–363] from case p = 2 to case 2 ⩽ p < ∞ .