Boundary behavior of solutions to a singular Dirichlet problem with a nonlinear convection

Li, Bo ; Zhang, Zhijun

Electronic Journal of Differential Equations, Tome 2015 (2015).

Résumé

Summary: 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.

Détail

Classification : 35J65, 35B05, 35J25, 60J50
Keywords: semilinear elliptic equation, singular Dirichlet problem, nonlinear convection term, classical solution, boundary behavior