Infinitely Many Sign-changing Solutions for a Schrdinger Equation in R^N

We consider the existence of infinitely many sign-changing solutions for the nonlinear time-independent schrodinger equations of the form where Vλ(x) =λa(x) + 1. This problem originates from various problems in physics and mathematical physics. In constructive field theory, (1.1) is called a nonlinear Euclidean scalar field equation. In chemical dynamics, a solution of (1.1) is a stationary state of the reaction diffusion

We consider the existence of infinitely many sign-changing solutions for the nonlinear timeindependent schroedinger equations of the form