Saddle solutions for bistable symmetric semilinear elliptic equations

This paper concerns the existence and asymptotic characterization of saddle solutions in ${mathbb {R}^{3}}$ for semilinear elliptic equations of the form $$-Delta u + W'(u) = 0,quad (x, y, z) in {mathbb {R}^{3}} qquadqquadqquad (0.1)$$ where ${W in mathcal{C}^{3}(mathbb {R})}$ is a double well symmetric potential, i.e. it satisfies W(?s) =  W(s) for ${s in mathbb {R},W(s) > 0}$ for ${s in (-1,1)}$ , ${W(pm 1) = 0}$ and ${W''(pm 1) > 0}$ . Denoted with ${theta_{2}}$ the saddle planar solution of (0.1), we show the existence of a unique solution ${theta_{3} in {mathcal{C}^{2}}(mathbb {R}^{3})}$ which is odd with respect to each variable, symmetric with respect to the diagonal planes, verifies ${0 < theta_{3}(x,y,z) < 1}$ for x, y, z >  0 and ${theta_{3}(x, y, z) to_{z to + infty} theta_{2}(x, y)}$ uniformly with respect to ${(x, y) in mathbb {R}^{2}}$ .