Existence and regularity of solutions to elliptic equation with singular convection term and lower order term. (25th June 2022) - He, Xiaohua; Huang, Shuibo; Tian, Qiaoyu
Abstract : In this paper, we consider the effects of singular convection term and lower order term on the existence and regularity of solutions to the following elliptic Dirichlet problem: − div M ( x ) ∇ u = − div u E ( x ) + f ( x ) u γ, x ∈ Ω, u ( x ) = 0, x ∈ ∂ Ω, $$ \left\{\begin{array}{rl}-\operatorname{div}\left(M(x)\nabla u\right)=-\operatorname{div}\left( uE(x)\right)+\frac{f(x)}{u^{\gamma }}, & x\in \Omega, \\ {}\kern45pt u(x)=0, & x\in \mathrm{\partial \Omega }, \end{array}\right. $$ where γ > 0, Ω ⊂ ℝ N ( N > 2 ) $$ \gamma >0, \Omega \subset {\mathbb{R}}^N\left(N>2\right) $$ is a bounded smooth domain with 0 ∈ Ω, f ∈ L m ( Ω ) $$ 0\in \Omega, f\in {L}^m\left(\Omega \right) $$ with m ≥ 1 $$ m\ge 1 $$ is a non‐negative function. The main results of this paper show the combined effects of singular convection term and lower order term on the existence and regularity of solution to this problem.