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

Abstract

Starting from a bound state (positive or sign-changing) solution to $$ -Δω_m =|ω_m|^{p-1} ω_m -ω_m \ \ \mbox{in}\ \R^n, \ ω_m \in H^2 (\R^n)$$ and solutions to the Helmholtz equation $$ Δu_0 + λu_0=0 \ \ \mbox{in} \ \R^n, \ λ>0, $$ we build new Dancer's type entire solutions to the nonlinear scalar equation $$ -Δu =|u|^{p-1} u-u \ \ \mbox{in} \ \R^{m+n}. $$

11 pages