Let $B(0;1)=\{x \in \mathbb{R}^N;|x|≤1\}$, the ball of $\mathbb{R}^N$ equipped with the euclidian scalar product $$x⋅y=x_1y_1+...+x_Ny_N,\ \ \ x=(x_1,...,x_N),\ \ \ y=(y_1,...,y_N)\ \ \ |x|=\sqrt{x \cdot x}$$ Let $\alpha$ and $\beta$ two real numbers, $\alpha \geq0$ , $a \in \partial B(0,1)$ fix. $$h(x)=\log\left((1−x\cdot a)^{\alpha}+|x−a|^{\beta}\right)$$ Can we find $\alpha$ and $\beta$ such that $h$ is harmonic in $B(0,1)$?
( harmonic signifies $\Delta_{x}h(x)=\displaystyle \sum_{i=0}^N \frac{\partial^2 h}{\partial x_i^2}$ ).