Saddle solutions for AllenCahn type equations involving the prescribed mean curvature operator
Abstract
The goal of this paper is to investigate the existence of saddle solutions for some classes of elliptic partial differential equations of the AllenCahn type, formulated as follows: \begin{equation*} div\left(\frac{\nabla u}{\sqrt{1+\nabla u^2}}\right) + A(x,y)V'(u)=0~~\text{ in }~~\mathbb{R}^2. \end{equation*} Here, the function $A:\mathbb{R}^2\to\mathbb{R}$ exhibits periodicity in all its arguments, while $V:\mathbb{R}\to\mathbb{R}$ characterizes a doublewell symmetric potential with minima at $t=\pm\alpha$.
 Publication:

arXiv eprints
 Pub Date:
 April 2024
 DOI:
 10.48550/arXiv.2404.11697
 arXiv:
 arXiv:2404.11697
 Bibcode:
 2024arXiv240411697I
 Keywords:

 Mathematics  Analysis of PDEs