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2007 Saddle-type solutions for a class of semilinear elliptic equations
Francesca Alessio, Alessandro Calamai, Piero Montecchiari
Adv. Differential Equations 12(4): 361-380 (2007).

Abstract

We consider a class of semilinear elliptic equations of the form \begin{equation}\label{eq:abs} -\Delta u(x,y)+W'(u(x,y))=0,\quad (x,y)\in\mathbb{R}^{2} \end{equation} where $W:\mathbb{R}\to\mathbb{R}$ is modeled on the classical two well Ginzburg-Landau potential $W(s)=(s^{2}-1)^{2}$. We show, via variational methods, that for any $j\geq 2$, the equation (\ref{eq:abs}) has a solution $v_{j}\in C^{2}(\mathbb{R}^{2})$ with $|v_{j}(x,y)|\leq 1$ for any $(x,y)\in\mathbb{R}^{2}$ satisfying the following symmetric and asymptotic conditions: setting $\tilde v_{j}(\rho,\theta)=v_{j}(\rho\cos( \theta),\rho\sin(\theta))$, there results \begin{eqnarray*} &\hbox{$\tilde v_{j}(\rho,\frac{\pi}{2}+\theta)=-\tilde v_{j}(\rho,\frac{\pi}{2}-\theta)$ and $\tilde v_{j}(\rho,\theta+\frac{\pi}{j})=-\tilde v_{j}(\rho,\theta)$, $\forall (\rho,\theta)\in \mathbb{R}^{+}\times\mathbb{R}$}&\\ & \hbox{and $\tilde v_{j}(\rho,\theta)\to 1$ as $\rho\to+\infty$ for any $\theta\in [\frac{\pi}2-\frac{\pi}{2j},\frac{\pi}2)$.}& \end{eqnarray*}

Citation

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Francesca Alessio. Alessandro Calamai. Piero Montecchiari. "Saddle-type solutions for a class of semilinear elliptic equations." Adv. Differential Equations 12 (4) 361 - 380, 2007.

Information

Published: 2007
First available in Project Euclid: 18 December 2012

zbMATH: 1193.35058
MathSciNet: MR2305872

Subjects:
Primary: 35J60
Secondary: 34B15, 34B40, 35B05, 35J20, 47J30

Rights: Copyright © 2007 Khayyam Publishing, Inc.

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Vol.12 • No. 4 • 2007
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