We consider a class of elliptic problems of a logistic type $$-div(|{ abla}_u|^{m-2}{ abla}_u);=;w(x)u^q;-;(a(x))^{frac{m}{2}};f(u)$$ in a bounded domain of $mathbf{R}^N$ with boundary $partialOmega$ of class $C^2$, $u|_{partialOmega};=;+{infty}$, $omega;in;L^{infty}(Omega)$, 0 < q < 1 and $a;{in};C^{alpha}(ar{Omega})$, $mathbf{R}^+$ is non-negative for some $alpha;in$ (0,1), where $mathbf{R}^+;=;[0,;infty)$. Under suitable growth assumptions on a, b and f, we show the exact blow-up rate and uniqueness of the large solutions. Our proof is based on the method of sub-supersolution.