带梯度项的半线性椭圆方程整体解的存在性

Existence of Entire Solutions for Semilinear Elliptic Problems with Convection Terms

中文关键词:  半线性椭圆方程  整体解  梯度项  存在性

英文关键词:semilinear elliptic equation  entire solution  convection term  existence

中文摘要:

      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.

英文摘要:

      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.