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
$$-\Delta u + W'(u) = 0,\quad (x, y, z) \in {\mathbb {R}^{3}} \qquad\qquad\qquad (0.1)$$
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 \({(x, y) \in \mathbb {R}^{2}}\) .

Mathematics Subject Classification (2010)

35J60 35B05 35B40 35J20 34C37 


Elliptic equations Variational methods Entire solutions 


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Copyright information

© Springer Basel 2012

Authors and Affiliations

  1. 1.Dipartimento di Ingegneria Industriale e Scienze MatematicheUniversità Politecnica delle MarcheAnconaItaly

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