Summary: We establish the existence, uniqueness and blow-up rate near the boundary of boundary blow-up solutions to \(p\)-Laplacian elliptic equations of logistic type \(-\Delta _{p}u = a(x)h(u) - b(x)f(u)\), where \(\Delta _{p}\)u = div\( (|\nabla u|^{ p - 2}\nabla u\)) with \(p > 1\), \(h(u)/u ^{ p - 1}\) is non-increasing and \(f(u)\) is a function whose variation at infinity may be regular or rapid. In particular, our result regarding the blow-up rate reveals the main difference between regular variation function \(f\) and rapid variation function \(f\).