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

We establish the uniqueness and the blow-up rate of the large positive solution of the quasi-linear elliptic problem −Δpu=λup−1−b(x)h(u) in BR(x0) with boundary condition u=+∞ on ∂BR(x0), where BR(x0) is a ball centered at x0∈RN 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)/up−1 increasing on (0,∞) and h(u)∼uq−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 −Δu=λu−b(x)h(u), J. Differential Equations 247 (2009) 344–363] from case p=2 to case 2⩽p<∞.