March/April 2012 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, K. Saoudi
Adv. Differential Equations 17(3/4): 369-400 (March/April 2012). DOI: 10.57262/ade/1355703090

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.

Citation

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R. Dhanya. J. Giacomoni. S. Prashanth. K. Saoudi. "Global bifurcation and local multiplicity results for elliptic equations with singular nonlinearity of super exponential growth in $\mathbb{R}^2$." Adv. Differential Equations 17 (3/4) 369 - 400, March/April 2012. https://doi.org/10.57262/ade/1355703090

Information

Published: March/April 2012
First available in Project Euclid: 17 December 2012

zbMATH: 1258.35073
MathSciNet: MR2919106
Digital Object Identifier: 10.57262/ade/1355703090

Subjects:
Primary: 35J20 , 35J65 , 35J70

Rights: Copyright © 2012 Khayyam Publishing, Inc.

Vol.17 • No. 3/4 • March/April 2012
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