Bifurcations of some elliptic problems with a singularnonlinearity via Morse index

Zongming Guo; Zhongyuan Liu; Juncheng Wei; Feng Zhou

doi:10.3934/cpaa.2011.10.507

Article Contents

2011, Volume 10, Issue 2: 507-525. Doi: 10.3934/cpaa.2011.10.507

This issue Previous Article A system of the Hamilton--Jacobi and the continuity equations in the vanishing viscosity limit Next Article Positive solutions for singular elliptic equations with critical Hardy-Sobolev exponent

Bifurcations of some elliptic problems with a singular nonlinearity via Morse index

1.
Department of Mathematics, Henan Normal University, Xinxiang, 453007
2.
Department of Mathematics, Chinese University of Hong Kong, Shatin, New Territories, Hong Kong
3.
Department of Mathematics, East China Normal University, Shanghai 200062

Received: January 2010

Revised: May 2010

Published: March 2011

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Abstract

We study the boundary value problem

$\Delta u=\lambda |x|^\alpha f(u)$ in $\Omega, u=1$ on $\partial \Omega\qquad$ (1)

where $\lambda>0$, $\alpha \geq 0$, $\Omega$ is a bounded smooth domain in $R^N$ ($N \geq 2$) containing $0$ and $f$ is a $C^1$ function satisfying $\lim_{s \to 0^+} s^p f(s)=1$. We show that for each $\alpha \geq 0$, there is a critical power $p_c (\alpha)>0$, which is decreasing in $\alpha$, such that the branch of positive solutions possesses infinitely many bifurcation points provided $p > p_c (\alpha)$ or $p > p_c (0)$, and this relies on the shape of the domain $\Omega$. We get some important estimates of the Morse index of the regular and singular solutions. Moreover, we also study the radial solution branch of the related problems in the unit ball. We find that the branch possesses infinitely many turning points provided that $p>p_c (\alpha)$ and the Morse index of any radial solution (regular or singular) in this branch is finite provided that $0 < p \leq p_c (\alpha)$. This implies that the structure of the radial solution branch of (1) changes for $0 < p \leq p_c (\alpha)$ and $p > p_c (\alpha)$.

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

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