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

Starting from a bound state (positive or sign-changing) solution to $$-\mathrm{\Delta }{\omega }_m=\mathrm{\mid }{\omega }_m{\mathrm{\mid }}^{p-1}{\omega }_m-{\omega }_{m}in{\mathbb{R}}^n,{\omega }_m\in H^2({\mathbb{R}}^n)$$ and solutions to the Helmholtz equation $$ \Delta u_0

• \lambda u_0=0 \ \ \mbox{in} \ \R^n, \ \lambda>0, $$webuildnewDance{r}^{\mathrm{\prime }}stypeentiresolutionstothenonlinearscalarequation$$ -\Delta u =|u|^{p-1} u-u \ \ \mbox{in} \ \R^{m+n}. $$