On the global branch of positive radial solutions of an elliptic problem with singular nonlinearity

Zongming Guo; Xuefei Bai

doi:10.3934/cpaa.2008.7.1091

Article Contents

2008, Volume 7, Issue 5: 1091-1107. Doi: 10.3934/cpaa.2008.7.1091

This issue Previous Article Hyperbolic balance laws with a dissipative non local source Next Article Positive solutions for critical elliptic systems in non-contractible domains

On the global branch of positive radial solutions of an elliptic problem with singular nonlinearity

Zongming Guo¹ and
Xuefei Bai^2,

1.
Department of Mathematics, Henan Normal University, Xinxiang, 453007
2.
Department of Mathematics, Dong Hua University, Shanghai, 200051

Received: August 31, 2007

Revised: January 31, 2008

Published: August 2008

Abstract Related Papers Cited by

Abstract

We consider the following problem

$\Delta u=\lambda [\frac{1}{u^p}-\frac{1}{u^q}]$ in $B$, $u=\kappa \in (0,(\frac{p-1}{q-1})^{-1/(p-q)} ]$ on $\partial B$, $0 < u < \kappa$

in $B$, where $p > q > 1$ and $B$ is the unit ball in $\mathbb R^N$ ($N \geq 2$). We show that there exists $\lambda_\star>0$ such that for $0<\lambda <\lambda_\star$, the maximal solution is the only positive radial solution. Furthermore, if $2 \leq N < 2+\frac{4}{p+1} (p+\sqrt{p^2+p})$, the branch of positive radial solutions must undergo infinitely many turning points as the maxima of the radial solutions on the branch go to 0. The key ingredient is the use of a monotonicity formula.

Keywords:

Global branch,
infinitely many turning points,
semilinear elliptic problems with singularity.

Mathematics Subject Classification: Primary: 35B45; Secondary: 35J40.

Citation:

Article Metrics

HTML views() PDF downloads(74) Cited by(0)

On the global branch of positive radial solutions of an elliptic problem with singular nonlinearity

Abstract

Access History

Article Metrics

Access History

Other Articles By Authors

Catalog

On the global branch of positive radial solutions of an elliptic problem with singular nonlinearity

Abstract

Access History

SHARE

Article Metrics

Access History

Other Articles By Authors

Catalog

Export File

Citation

Format

Content