Blow-up rates and uniqueness of entire large solutions to a semilinear elliptic equation with nonlinear convection term

Bo Li,Haitao Wan

doi:10.1186/s13661-018-1101-0

Abstract

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}, xin mathbb{R}^{N} (Ngeq 3), where mu > 0, q > 0 and a, bin 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 fin 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.

Highlights

1 Introduction and the main results The purpose of this paper is to study the blow-up rates and uniqueness of entire large solutions to the following elliptic equation: u = a(x)f (u) + μb(x)|∇u|q, x ∈ RN, u(x) > 0, (1.1)
When Ω is a bounded domain with a C2-boundary, b ≡ 1 in Ω ⊆ RN (N = 2) and f (u) = eu; Bieberbach [5] first studied the existence, uniqueness and asymptotic behavior of large solutions to Eq (1.1)
Motivated by the work of [2] and [47,48,49], in this paper, we determine the exact asymptotic behavior of entire large solutions to Eq (1.1) at infinity in RN, and we show that the convection term μb(x)|∇u(x)|q does not affect the asymptotic behavior under certain conditions

Summary

Introduction

When Ω is a bounded domain with a C2-boundary, b ≡ 1 in Ω ⊆ RN (N = 2) and f (u) = eu; Bieberbach [5] first studied the existence, uniqueness and asymptotic behavior of large solutions to Eq (1.1). Keller [26] and Osserman [39] conducted systematic research on Eq (1.1) and obtained the following important results: (i) If Ω ⊆ RN is a bounded domain, b ≡ 1 on Ω , and f satisfies (f1), Eq (1.1) has a classical large solution if and only if the Keller–Osserman condition (f2) holds.

Objectives

Results

Conclusion