On the Helmholtz equation and Dancer’s-type entire solutions for nonlinear elliptic equations

Abstract:

Starting from a bound state (positive or sign-changing) solution to \begin{equation*} -\Delta \omega _m =|\omega _m|^{p-1} \omega _m -\omega _m \ \ \text {in}\ \mathbb {R}^m, \end{equation*} and solutions to the Helmholtz equation \begin{equation*} \Delta u_0 + \lambda u_0=0 \ \ \text {in} \ \mathbb {R}^n, \ \lambda >0, \end{equation*} we build new Dancer’s-type entire solutions to the nonlinear scalar equation \begin{equation*} -\Delta u =|u|^{p-1} u-u \ \ \text {in} \ \mathbb {R}^{m+n}. \end{equation*}