Saddle-type solutions for a class of semilinear elliptic equations

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*}