boundary behavior of solutions to a singular dirichlet problem with a nonlinear convection

boundary behavior of solutions to a singular dirichlet problem with a nonlinear convection

;Bo Li;Zhijun Zhang
icsoft 2006 - 1st international conference on software and data technologies, proceedings 2015 Vol. 2015 pp. 1-18
68
li2015electronicboundary

Abstract

In this article we analyze the exact boundary behavior of solutions to the singular nonlinear Dirichlet problem $$\displaylines{ -\Delta u=b(x)g(u)+\lambda|\nabla u|^q+\sigma, \quad u>0, \; x \in \Omega,\cr u\big|_{\partial \Omega}=0, }$$ where $\Omega$ is a bounded domain with smooth boundary in $\mathbb{R}^N$, $q\in (0, 2]$, $\sigma>0$, $\lambda> 0$, $g\in C^1((0,\infty), (0,\infty))$, $\lim_{s \to 0^+}g(s)=\infty$, $g$ is decreasing on $(0, s_0)$ for some $s_0>0$, $b \in C_{\rm loc}^{\alpha}({\Omega})$ for some $\alpha\in (0, 1)$, is positive in $\Omega$, but may be vanishing or singular on the boundary. We show that $\lambda |\nabla u|^q$ does not affect the first expansion of classical solutions near the boundary.

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