Abundance of entire solutions to nonlinear elliptic equations by the variational method

Abstract

We study entire bounded solutions to the equation $\Delta u-u+u^3=0$ in ${\mathbb{R}}^2$. Our approach is purely variational and is based on concentration arguments and symmetry considerations. This method allows us to construct in a unified way several types of solutions with various symmetries (radial, breather type, rectangular, triangular, hexagonal, etc.), both positive and sign-changing. The method is also applicable for more general equations in any dimension.

30 pages, 28 figures