Infinitely Many Sign-Changing Solutions for a SchröDinger Equation With Competing Potentials

Consider the following nonlinear problem with competing potentials: -Delta u+P(|y|)u=Q(|y|)|u|p-1u,in & Ropf;N$$ -\Delta u+P\left(|y|\right)u=Q\left(|y|\right){\left|u\right|}<^>{p-1}u,\mathrm{in}\kern0.3em {\mathbb{R}}<^>N $$, where N >= 3,1<p<N+2N-2$$ N\ge 3,\kern0.3em 1<p<\frac{N+2}{N-2} $$, and P(|y|),Q(|y|)$$ P\left(|y|\right),\kern0.3em Q\left(|y|\right) $$ are positive functions. We construct infinitely many nonradial sign-changing solutions to the equation above by the Lyapunov-Schmidt reduction method.