### 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.

$f \in C((0,1)\times (0,\infty),\mathbb{R})$ 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.

#### Keywords

Elliptic equations; positive solutions; radial symmetry; Riemannian isometric; the Hénon equation

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