Saddle solutions for bistable symmetric semilinear elliptic equations

This paper concerns the existence and the asymptotic characterization of saddle solutions in $R^{3}$ for semilinear elliptic equations of the form egin{equation}label{eq:abs} -Delta u+W'(u)=0,quad (x,y,z)inR^{3} end{equation} where $WinCC^{3}(R)$ is a double well symmetric potential, i.e. it satisfies $W(-s)=W(s)$ for $sinR$, $W(s)> 0$ for $sin (-1,1)$, $W(pm 1)=0$ and $W''(pm 1)>0$. Denoted with $ heta_{2}$ the saddle planar solution of ( ef{eq:abs}), we show the existence of a unique solution $ heta_{3}in C^{2}(R^{3})$ which is odd with respect to each variable, symmetric with respect to the diagonal planes, verifies $0< heta_{3}(x,y,z)<1$ for $x,y,z>0$ and $ heta_{3}(x,y,z) o_{z o+infty} heta_{2}(x,y)$ uniformly with respect to $(x,y)inR^{2}$.