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

Abstract

We show that each positive solution of ∆u + f (|x|, u) = 0 in D, u = 0 on ∂D which satisfies u(r, θ) = u(r, θ + 2π/n) by the polar coordinates is radially symmetric and ur(|x|) < 0 for each r = |x| ∈ (0, 1). We apply our result to the Hénon equation.

Symmetry-breaking bifurcation for the one-dimensional Hénon equation

“…Badiale and Serra [2] found at least N/2 − 1 positive non-radial solutions of (1.2) when N ≥ 4, 1 < p < (N + 1)/(N − 3) and l > 0 is sufficiently large, where x denote the largest integer less than or equal to a real number x. When N = 2, Shioji and Watanabe [13] proved the following: for each l > 0, if p > 1 is sufficiently large, then (1.2) has l/2 positive non-radial solutions, where x denote the smallest integer greater than or equal to x; for each p > 1, the number of positive non-radial solutions of (1.2) tends to infinity as l → ∞. When N = 3 and 1 < p < 5, Shioji [12] showed that the number of positive non-radial solutions of (1.2) tends to infinity as l → ∞.…”