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

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

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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: September 2007

Revised: February 2008

Published: September 2008

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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.

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