I want to show that the following Dirichlet problem \begin{equation} \begin{cases} \Delta u = 0 & \text{in } \Omega,\\ u = g & \text{on }\partial \Omega, \end{cases} \end{equation} where $\Omega = (0,1)^2$ and $g(x, y) = 4xy - 7e^x$, has a unique solution.
I clearly only have to show the existence as uniqueness is given by the maximum principle. In my functional analysis course, we've seen that such a problem admits a solution if and only if each point of the boundary admits a barrier, i.e. $$\forall (x_0, y_0) \in \partial \Omega, \quad \exists w \in C(\bar\Omega) ~~\text{with }~ w(x_0, y_0) = 0~~ \text{and}~~ w > 0 ~~ \text{on }~\bar\Omega\backslash \{x_0, y_0\},$$ and $w$ is superharmonic. Therefore I have to find $w$ defined on the square $\Omega$ that satisfies the above conditions. The only functions I can think of have the form $$w(x, y) = |x - x_0| + |y - y_0|$$ but it doesn't seem to be superharmonic.. Any help ?