Existence of Entire Solutions for Semilinear Elliptic Problems with Convection Terms

DOI：10.3770/j.issn:2095-2651.2015.04.007

 作者 单位 薛洪涛 烟台南山学院数学物理教学部, 山东 烟台 265713

应用上下解方法、摄动方法和椭圆型偏微分方程的估计理论等,本文指出半线性椭圆问题$- \Delta u +a(x)|\nabla u|^q=\lambda b(x)g(u)$, $u>0$, $x\in \mathbb R^N$, $\lim_{|x|\rightarrow \infty} u(x)=0$至少存在一个解,其中$10$, $a$ 和$b$ 均为局部 H\"{o}lder 连续函数, 且对任意的$x\in \mathbb R^N$，有$a\geq 0$, $b>0$, 函数$g\in C^1((0,\infty), (0,\infty))$且可能在零点具有奇异性,在无穷远处无界.

By a sub-supersolution method and a perturbed argument, we show the existence of entire solutions for the semilinear elliptic problem $- \Delta u +a(x)|\nabla u|^q=\lambda b(x)g(u)$, $u>0$, $x\in \mathbb R^N$, $\lim_{|x|\rightarrow \infty} u(x)=0$, where $q\in (1,2]$, $\lambda>0$, $a$ and $b$ are locally H\"{o}lder continuous, $a\geq 0$, $b>0$, $\forall x\in \mathbb R^N$, and $g\in C^1((0,\infty), (0,\infty))$ which may be both possibly singular at zero and strongly unbounded at infinity.