Saddle-type solutions for a class of semilinear elliptic equations
Saddle-type solutions for a class of semilinear elliptic equations
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 …