the existence of solution of Laplace equation

Question

$\Omega=\mathbb{D}^{\star},\mathbb{D}^{\star}$ is open ball in $\mathbb{R}^2$without zero. is there a solution $u\in C^2(\Omega)\cap C(\overline{\Omega})$ of $\vartriangle u(x)=0,x\in\Omega,u(0)=1,u(x)=0,\|x\|=1$?

Answer 1

No. From the Maximum principle, solutions to Laplace's Equation must obtain their maximums and minimums along the boundary of their domain. So any solution that is 0 along the boundary must be 0 over the entire domain, which contradicts your condition of $u(0) = 1.$