带有平均曲率算子的拟线性问题在无穷区间上的可解性

Solvability of Quasilinear Problems with Mean Curvature Operator on Infinite Interval

摘要: 运用Leray-Schauder原理讨论无穷区间上带有平均曲率算子的拟线性问题。$\{\begin{array}{l}{\left(\phi \left(u^{\prime }\right)\right)}^{\prime }=f\left(t,u\left(t\right),u^{\prime }\left(t\right)\right),t\in \left[0,\infty \right),\\ \underset{t\to \infty }{\lim }u\left(t\right)=0,\underset{t\to \infty }{\lim }{u}^{\prime }\left(t\right){\text{e}}^t=0,\end{array}$ (1)具有可解性，其中$f:\left[0,\infty \right)\times ℝ\times \left(-1,1\right)\to ℝ$ 连续。$\phi \left(s\right)=\frac{s}{\sqrt{1-s^2}},s\in \left(-1,1\right)$ 。

Abstract: In this paper, by using the Leray-Schauder theory, we are concerned with the solvability of $\{\begin{array}{l}{\left(\phi \left(u^{\prime }\right)\right)}^{\prime }=f\left(t,u\left(t\right),u^{\prime }\left(t\right)\right),t\in \left[0,\infty \right),\\ \underset{t\to \infty }{\lim }u\left(t\right)=0,\underset{t\to \infty }{\lim }{u}^{\prime }\left(t\right){\text{e}}^t=0,\end{array}$ where $f:\left[0,\infty \right)\times ℝ\times \left(-1,1\right)\to ℝ$ is continuous, $\phi \left(s\right)=\frac{s}{\sqrt{1-s^2}},s\in \left(-1,1\right)$ .