Advances in Differential Equations

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

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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.


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