For which real constants is the function $$ v(x):=x_1^3+kx_1x_2^2 $$ harmonic on $\mathbb{R}^n$?
To my calculation, the equation $$ \Delta v=\sum_{k=1}^n\frac{\partial^2 v}{\partial x_k^2}=0 $$ is fullfilled when $$ x_1=0\vee k=-3. $$
What does this mean for the question that I have to answer here? Is it just: "For $k=-3$ the function is harmonic."? Or do I have to add: When $x_1=0$, it follows that $v\equiv 0$ and then every $k\in\mathbb{R}$ makes $v$ harmonic?