Bifurcations of some elliptic problems with a singularnonlinearity via Morse index
Zongming Guo, Zhongyuan Liu, Juncheng Wei, Feng Zhou · Communications on Pure & Applied Analysis · 2011
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 smoothdomain in $R^N$ ($N \geq 2$) containing $0$ and $f$ is a $C^1$ functionsatisfying$\lim_{s \to 0^+} s^p f(s)=1$. We show that for each $\alpha \geq 0$,there is a criticalpower $p_c (\alpha)>0$, which isdecreasing in $\alpha$, such that the branch of positivesolutions possesses infinitely many bifurcationpoints provided $p > p_c (\alpha)$ or $p > p_c (0)$, and this relies onthe shape of the domain $\Omega$. We get some important estimatesof the Morse index of the regular andsingular solutions. Moreover, we also study the radial solutionbranch of the related problems in the unit ball. We find that thebranch possesses infinitely many turning points provided that$p>p_c (\alpha)$ and the Morse index of any radial solution (regular orsingular) in this branch is finite provided that$0 p_c (\alpha)$.