Radial symmetry of $n$-mode positive solutions for semilinear elliptic equations in a disc and its application to the Hénon equation

Abstract

Let $f \in C((0,1)\times (0,\infty),\mathbb{R})$ and $n \in \mathbb{N}$ with $n \geq 2$ such that for each $u \in (0,\infty)$, $r\mapsto r^{2-2n}f(r,u)\colon (0,1)\rightarrow \mathbb{R}$ is nonincreasing and let $D=\{x=(x_1,x_2)\in\mathbb{R}^2: |x|< 1\}$. We show that each positive solution of $$ \Delta u + f(|x|,u) =0 \quad\text{in $D$,} \qquad u=0 \quad\text{on $\partial D$} $$ which satisfies $u(r,\theta)= u(r,\theta+2\pi/n)$ by the polar coordinates is radially symmetric and $u_r(|x|)< 0$ for each $r=|x| \in (0,1)$. We apply our result to the Hénon equation.