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September/October 2013 Multiplicity and singular solutions for a Liouville type system in a ball
Juan Dávila, Isabel Flores, Ignacio Guerra
Adv. Differential Equations 18(9/10): 797-824 (September/October 2013).

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 $\mathcal S$.

Citation

Download Citation

Juan Dávila. Isabel Flores. Ignacio Guerra. "Multiplicity and singular solutions for a Liouville type system in a ball." Adv. Differential Equations 18 (9/10) 797 - 824, September/October 2013.

Information

Published: September/October 2013
First available in Project Euclid: 2 July 2013

zbMATH: 1278.35083
MathSciNet: MR3100052

Subjects:
Primary: 35B07 , 35J47 , 35J57 , 35J60

Rights: Copyright © 2013 Khayyam Publishing, Inc.

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Vol.18 • No. 9/10 • September/October 2013
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