Advances in Differential Equations

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

Juan Dávila, Isabel Flores, and Ignacio Guerra

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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 $\mathcal S$.

Article information

Adv. Differential Equations, Volume 18, Number 9/10 (2013), 797-824.

First available in Project Euclid: 2 July 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35J60: Nonlinear elliptic equations 35J47: Second-order elliptic systems 35J57: Boundary value problems for second-order elliptic systems 35B07: Axially symmetric solutions


Dávila, Juan; Flores, Isabel; Guerra, Ignacio. Multiplicity and singular solutions for a Liouville type system in a ball. Adv. Differential Equations 18 (2013), no. 9/10, 797--824.

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