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In this paper, we analyze the blow-up rates and uniqueness of entire large solutions to the equation $\Delta u=a(x)f(u)+ \mu b(x) |\nabla u|^{q}$ , $x\in \mathbb{R}^{N}$ ( $N\geq 3$ ), where $\mu > 0$ , $q > 0$ and $a, b\in \mathrm {C}^{\alpha }_{\mathrm{loc}}(\mathbb{R}^{N})$ ( $\alpha \in (0, 1)$ ). The weight a is nonnegative, b is able to change sign in $\mathbb{R}^{N}$ , and $f\in C^{1}[0, \infty )$ is positive and nondecreasing on $(0, \infty )$ and rapidly or regularly varying at infinity. Additionally, we investigate the uniqueness of entire large solutions.

We here investigate the existence and uniqueness of the nontrivial, nonnegative solutions of a nonlinear ordinary differential equation: satisfying a specific decay rate: with and . Here and . Such a solution arises naturally when we study a very singular self-similar solution for a degenerate parabolic equation with nonlinear convection term defined on the half line .