Condition of $u$ being trivial when harmonic

Question

If $\Delta u = 0$ in $\mathbb{R}^n$ and $u \in L^p$ for some $p \in (1,\infty)$, then $u \equiv 0$.

I tried to use the maximum principle or something like that. Only I couldn't. Any tips?

Answer 1

Let $r>0$, by the mean value property, we have $$ |u(x)| \leq \frac 1 {r^2 \pi} \int_{B_r(x)} |u(y)| \ d y. $$ Using Hoelder, we have $$ \frac 1 {r^2 \pi} \int_{B_r(x)} |u(y)| \ d y \leq \frac 1 {\pi^{1-1/q}} r^{-2 + 2/q} ||u ||_{L^p(\mathbb{R}^2)}. $$ Where $1/q + 1/p =1$. Hence $-2+2/q = -2/p$. Then send $r \to \infty$.