Analytic global bifurcation and infinite turning points for very singular problems

Abstract

Let \(\Omega \) be a domain in \(\mathbb {R}^N, N\ge 2\), \(\lambda >0\) a bifurcation parameter and \(f: \mathbb {R}\rightarrow \mathbb {R}\) a “real analytic” type map such that \(f(t)\) has superlinear growth as \(t \rightarrow \infty \). We consider semilinear elliptic PDEs with the presence of a strong singular term as below: $$\begin{aligned} ( P_\lambda )\left\{ \begin{array}{ll} - \Delta u= \lambda (u^{-\delta }+ f(u)) \quad \text{ in } \,\Omega ,\\ u> 0\quad \text{ in } \,\Omega , \;\;u\vert _{\partial \Omega } = 0. \end{array} \right. \end{aligned}$$ Here the singular exponent \(\delta \) is allowed to be any positive number. We are interested in this work to analyse the problem \((P_\lambda )\) using the framework of analytic bifurcation theory as developed in the works of Buffoni et al. (Arch. Ration. Mech. Anal. 152:207–240, 2000, Arch. Ration. Mech. Anal. 152:241–271, 2000). We obtain an analytic global unbounded path of solutions to \((P_\lambda )\) for any \(\delta >0\) using this framework. In two dimensions when \(0<\delta <1\) and for certain classes of nonlinearities \(f\) that have critical growth (in the sense of Trudinger-Moser imbedding), we show the existence of an analytic unbounded path of solutions to \((P_\lambda )\) whose Morse index is unbounded along this path. As a consequence, this path admits infinitely many “turning points”.