Advances in Differential Equations

Saddle-type solutions for a class of semilinear elliptic equations

Francesca Alessio, Alessandro Calamai, and Piero Montecchiari

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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*}

Article information

Source
Adv. Differential Equations, Volume 12, Number 4 (2007), 361-380.

Dates
First available in Project Euclid: 18 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.ade/1355867455

Mathematical Reviews number (MathSciNet)
MR2305872

Zentralblatt MATH identifier
1193.35058

Subjects
Primary: 35J60: Nonlinear elliptic equations
Secondary: 34B15: Nonlinear boundary value problems 34B40: Boundary value problems on infinite intervals 35B05: Oscillation, zeros of solutions, mean value theorems, etc. 35J20: Variational methods for second-order elliptic equations 47J30: Variational methods [See also 58Exx]

Citation

Alessio, Francesca; Calamai, Alessandro; Montecchiari, Piero. Saddle-type solutions for a class of semilinear elliptic equations. Adv. Differential Equations 12 (2007), no. 4, 361--380. https://projecteuclid.org/euclid.ade/1355867455


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