Radial symmetry of $n$-mode positive solutions for semilinear elliptic equations in a disc and its application to the Hénon equation - Shioji, Naoki; Watanabe, Kohtaro (Topological Methods in Nonlinear Analysis, 2016)
Let $f \in C((0,1)× (0,∞),\mathbb{R})$ and $n \in \mathbb{N}$ with $n ≥ 2$such that for each $u \in (0,∞)$,$r\mapsto r^{2-2n}f(r,u)\colon (0,1)→ \mathbb{R}$ is nonincreasing and let $D=\{x=(x_1,x_2)\in\mathbb{R}^2: |x|< 1\}$.We show that each positive solution of$$Δ u + f(|x|,u) =0 \quad\text{in $D$,}\qquad u=0 \quad\text{on $\partial D$}$$which satisfies$u(r,θ)= u(r,θ+2π/n)$ by the polar coordinatesis radially symmetric and $u_r(|x|)< 0$ for each $r=|x| \in (0,1)$.We apply our result to the Hénon equation.