Multiplicity and singular solutions for a Liouville type system in a ball

Abstract

We consider the Liouville system \begin{align*} -\Delta u = \lambda e^v , \quad -\Delta v = \mu e^u \quad \text{in } B \end{align*} with $u=v=0$ on $\partial B$, where $B$ is the unit ball in $\mathbb R^N$, $N\geq 3$, and $\lambda$ and $\mu$ are positive parameters. First we show that radial solutions in $B \setminus \{0\}$ are either regular or have a $\log$-type singularity. Then, in dimensions $3 \leq N \leq 9$ we prove that there is an unbounded curve $\mathcal S \subset (0,\infty)^2$ such that for each $(\mu,\lambda) \in\mathcal S$ there exist infinitely many regular solutions. Moreover, the number of regular solutions tends to infinity as $(\mu,\lambda)$ approaches a fixed point in $\mathcal S$.