On Helmholtz equation and Dancer's type entire solutions for nonlinear elliptic equations

TL;DR

This paper constructs new entire solutions to a nonlinear elliptic equation by combining bound state solutions and Helmholtz equation solutions, extending the class of known solutions.

Contribution

It introduces a method to generate Dancer's type solutions for nonlinear elliptic equations using solutions of related linear and nonlinear equations.

Findings

Constructed new Dancer's type solutions for nonlinear elliptic equations.

Extended the class of known solutions using bound state and Helmholtz solutions.

Provided a framework for generating solutions in higher dimensions.

Abstract

Starting from a bound state (positive or sign-changing) solution to $$ -\Delta \omega_m =|\omega_m|^{p-1} \omega_m -\omega_m \ \ \mbox{in}\ \R^n, \ \omega_m \in H^2 (\R^n)$$ and solutions to the Helmholtz equation $$ \Delta u_0 + \lambda u_0=0 \ \ \mbox{in} \ \R^n, \ \lambda>0, $$ we build new Dancer's type entire solutions to the nonlinear scalar equation $$ -\Delta u =|u|^{p-1} u-u \ \ \mbox{in} \ \R^{m+n}. $$