Nonradial entire solutions for Liouville systems

Abstract

We consider the following system of Liouville equations:We show the existence of at least n − n 3 global branches of nonradial solutions bifurcating from u1(x) = u2(x) = U (x) = log 64 (2 + µ) (8 + |x| 2 ) 2 at the values µ = −2 n 2 + n − 2 n 2 + n + 2 for any n ∈ N.

Multiple Axially Asymmetric Solutions to a Mean Field Equation on $\mathbb{S}^{2}$

“…To prove Theorem 1.2, we need to introduce the following lemma, which is slightly different from Lemma A.3 in [2].…”

“…We note that all the improper integrals are well defined due to the behavior of P m n (y) andP m n (y) at y = ±1 and the simplicity of the zeros of P m n (y) in (−1, 1) and the oddness of P 0 n (y) and evenness of P m n (y) when n, m are odd. See Lemma A.3 of [2] for a detailed explanation. Proof of Theorem 1.2.…”

“…The main difficulty is that the kernel of the nature associated operator is not one dimensional. In order to overcome this obstacle, we follow the method from [2] by searching for spaces with some symmetry. Multiple two-dimensional solutions of mean field equation on flat tori bifurcating from trivial solution can be seen in [10].…”

“…we have that Q∂2 u,u T(ρ n , 0)[w 0 , w 0 ] = 0. Hence(∂ u T(ρ n , 0)) −1 (I − Q)∂ 2 u,u T(ρ n , 0)[w 0 , w 0 ] = (∂ u T(ρ n , 0)) −1 ∂ 2 u,u T(ρ n , 0)[w 0 , w 0 ],and we denote this term as φ.…”