Global bifurcation and local multiplicity results for elliptic equations with singular nonlinearity of super exponential growth in $\mathbb{R}^2$

Abstract

In this paper, we study the solutions to the following singular elliptic problem of exponential type growth posed in a bounded smooth domain $\Omega \subset {{\mathbb R}}^2$: \begin{eqnarray*}( P_\lambda)\qquad \left\{\begin{array} {ll} & - \Delta u = \lambda (u^{-\delta}+h(u)e^{u^\alpha}) \quad\mbox{ in }\,\Omega,\\ & u> 0\quad\mbox{ in }\,\Omega, \;\;u\vert_{\partial\Omega} = 0. \end{array}\right. \end{eqnarray*} Here, $1\leq\alpha\leq 2$, $0<\delta<3$, ${\lambda}\geq 0$ and $h(t)$ is assumed to be a smooth ``perturbation" of $e^{t^{\alpha}}$ as $t \rightarrow \infty$ (see ${\bf (H1)-(H2)}$ below). We show the existence of an unbounded connected branch of solutions to $(P_{\lambda})$ emanating from the trivial solution at ${\lambda}=0$. In the radial case (i.e., when $\Omega =B_1$ and $u$ is radially symmetric) we make a detailed study of the blow-up/convergence of the solution branch as it approaches the asymptotic bifurcation point at infinity. In the critical case $\alpha=2$, we interpret the multiplicity results in terms of the corresponding bifurcation diagrams and the asymptotic profile of large solutions along the branch at infinity.