I want to ask whether my proof works for a theorem proved in Gidas-Ni-Nirenberg's second paper in 1981 (MR0634248), which is much shorter than the authors' proof. Many thanks for any discussion.
Theorem 1' (Page 378)
Let $u > 0$ be a $C^2$ solution of $-\Delta u = g(u)$ in $\mathbb{R}^n, n \geq 3$ with
$u(x) = O(|x|^{-m})$ at infinity, $m>0$
Assume (i) $g(s) \geq 0$ on $ 0\leq s \leq \|u\|_\infty$ and $g = g_1 + g_2$ with $g_1$ is Lipschitz and $g_2$ is continuous and non-decreasing, (ii) $g(s) = O(s^\alpha)$ near $s=0$ for some $\alpha > \frac{n+1}{m}$.
Then $u$ is radially symmetric about some point in $\mathbb{R}^n$ and $u_r < 0$ where $r$ is the radial coorinate about that point.
They shows the symmetry in $x_1$-axis by the well-known moving plane method. The first step is to show the set $\Lambda := \{ \lambda \in \mathbb{R}: u(x) > u(x^\lambda), \forall x \in \Sigma_\lambda\}$ contains $[R,\infty)$ for some large $R>0$.
Notations: $x^\lambda$ is the reflection of $x$ with respect to the plane $\{ x_1 = \lambda \}$. $\Sigma_\lambda = \{ x: x_1 < \lambda\}$.
To obtain such result, they need to show $\int_{\mathbb{R}^n} g(u(y)) dy > 0$. The authors used 2 pages to prove such thing, which is much longer than mine. (That's why I confused.)
My shorter proof:
Assume not, then $g(u(y)) = 0$ in $\mathbb{R}^n$ by $g \geq 0$. So $u$ is harmonic and bounded. The Liouville's theorem and decay condition implies $u \equiv 0$, which contradicts to the positivity of $u$.