Dirichlet problems for singular elliptic equations. II

Abstract:

Consider an elliptic equation \[ ( \ast )\quad L[u] = \sum \limits _{i,j = 1}^n {{a_{ij}}\frac {{{\partial ^2}u}}{{\partial {x_i}\partial {x_j}}} + \sum \limits _{i = 1}^n {{b_i}\frac {{\partial u}}{{\partial {x_i}}} + cu = f} } \] in a bounded domain $G$ in the half space ${x_n} > 0$ with boundary $\partial G = {S_1} \cup {S_2}$ of class ${C^{2 + \alpha }}$ where ${S_1}$ is contained in the hyperplane ${x_n} = 0$ and ${S_2}$ lies entirely in ${x_n} > 0$. The coefficient ${b_n}$ possesses certain type of singularity at ${x_n} = 0$. Let ${b_n} = h/k$ where $h \in {C^\alpha }(\bar G)$ and $k \to 0$ as ${x_n} \to 0$. It is found that the solvability of the Dirichlet problem of $L[u] = f$ in $G$ depends on the nature of singularity of ${b_n}$ and also the value of $h$ at ${x_n} = 0$.