# Nonradial entire solutions for Liouville systems

## TL;DR

This paper establishes the existence of multiple nonradial solutions for a Liouville system in the plane, demonstrating bifurcation phenomena at specific parameter values and expanding understanding of solution structures.

## Contribution

It introduces new nonradial solutions for Liouville systems and identifies bifurcation points, advancing the analysis of solution multiplicity in nonlinear PDEs.

## Key findings

- Existence of multiple nonradial solutions bifurcating from radial solutions.
- Identification of bifurcation points at specific parameter values.
- Construction of global solution branches for the Liouville system.

## Abstract

We consider the following system of Liouville equations: $$\left\{\begin{array}{ll}-\Delta u_1=2e^{u_1}+\mu e^{u_2}&\text{in }\mathbb R^2\\-\Delta u_2=\mu e^{u_1}+2e^{u_2}&\text{in }\mathbb R^2\\\int_{\mathbb R^2}e^{u_1}<+\infty,\int_{\mathbb R^2}e^{u_2}<+\infty\end{array}\right.$$ 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 $n\in\mathbb N$.