Nonradial entire solutions for Liouville systems

Abstract

We consider the following system of Liouville equations: $$\{\begin{array}{ll}-\Delta u_1=2e^{u_1}+\mu e^{u_2}& \text{in }{\mathbb{R}}^2\\ -\Delta u_2=\mu e^{u_1}+2e^{u_2}& \text{in }{\mathbb{R}}^2\\ {\int }_{{\mathbb{R}}^2}{e}^{u_1}<+\infty ,{\int }_{{\mathbb{R}}^2}{e}^{u_2}<+\infty & \end{array}$$ We show existence of at least $n-\left[\frac{n}{3}\right]$ global branches of nonradial solutions bifurcating from $u_1(x)=u_2(x)=U(x)=\log \frac{64}{(2+\mu ){\left(8+|x{|}^2\right)}^2}$ at the values $\mu =-2\frac{n^2+n-2}{n^2+n+2}$ for any $n\in \mathbb{N}$.

18 pages, accepted on Journal of Differential Equations

References in corpus (1)

• On non-topological solutions for planar Liouville Systems of Toda-type