@inproceedings{590ac66b9ad443cb833f6a8b9ce27aa5,

title = "New entire solutions to some classical semilinear elliptic problems",

abstract = "This paper deals with the construction of solutions to autonomous semilinear elliptic equations considered in entire space. In the absence of space dependence or explicit geometries of the ambient space, the point is to unveil internal mechanisms of the equation that trigger the presence of families of solutions with interesting concentration patterns. We discuss the connection between minimal surface theory and entire solutions of the Allen-Cahn equation. In particular, for dimensions 9 or higher, we build an example that provides a negative answer to a celebrated question by De Giorgi for this problem. We will also discuss related results for the (actually more delicate) standing wave problem in nonlinear Schr{\"o}dinger equations and for sign-changing solutions of the Yamabe equation.",

keywords = "Allen-Cahn equation, Standing waves for NLS, Yamabe equation",

author = "{Del Pino}, Manuel",

year = "2011",

month = aug,

day = "6",

language = "English",

isbn = "9789814324304",

series = "Proceedings of the International Congress of Mathematicians 2010, ICM 2010",

publisher = "World Scientific Publishing",

pages = "1934--1957",

editor = "R. Bhatia and A. Pal and G. Rangarajan and V. Srinivas and M. Vanninathan",

booktitle = "Proceedings of the International Congress of Mathematicians 2010, ICM 2010 (4 vol set)",

note = "International Congress of Mathematicians 2010, ICM 2010 ; Conference date: 19-08-2010 Through 27-08-2010",

}