Asymptotic behavior and uniqueness of entire large solutions to a quasilinear elliptic equation

Abstract

In this paper, combining the upper and lower solution method with perturbation theory, we study the asymptotic behavior of entire large solutions to Eq. $\Delta_{p}u=b(x)f(u),\,u(x)>0,\,x\in\mathbb{R},$ where $b\in C^{\alpha}_{\rm loc}(\mathbb{R}^{N})$ $(\alpha\in(0, 1))$ is positive in $\mathbb{R}^{N}$ $(N \geq 3)$, $f\in C^{1}[0, \infty)$ is positive on $(0, \infty)$ which satisfies a generalized Keller-Osserman condition and is rapidly varying or regularly varying with index $\mu\geq p-1$. We then discuss the uniqueness of solutions by the asymptotic behavior of entire large solutions at infinity.

Keywords

• $p$-laplacian
• asymptotic behavior
• entire large solutions