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

Naoki Shioji, Kohtaro Watanabe

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.

Keywords


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

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