## Advances in Differential Equations

- Adv. Differential Equations
- Volume 17, Number 3/4 (2012), 369-400.

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

R. Dhanya, J. Giacomoni, S. Prashanth, and K. Saoudi

#### 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.

#### Article information

**Source**

Adv. Differential Equations Volume 17, Number 3/4 (2012), 369-400.

**Dates**

First available in Project Euclid: 17 December 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.ade/1355703090

**Mathematical Reviews number (MathSciNet)**

MR2919106

**Zentralblatt MATH identifier**

1258.35073

**Subjects**

Primary: 35J65: Nonlinear boundary value problems for linear elliptic equations 35J20: Variational methods for second-order elliptic equations 35J70: Degenerate elliptic equations

#### Citation

Dhanya, R.; Giacomoni, J.; Prashanth, S.; Saoudi, K. Global bifurcation and local multiplicity results for elliptic equations with singular nonlinearity of super exponential growth in $\mathbb{R}^2$. Adv. Differential Equations 17 (2012), no. 3/4, 369--400. https://projecteuclid.org/euclid.ade/1355703090