Sign-changing solutions for schrödinger equations with vanishing and sign-changing potentials

In this article, we study the existence of sign-changing solutions for the following Schrödinger equation −Δu+λV(x)u=K(x)|u|p−2u     x∈ℝN,     u→0 as |x| →+∞−Δu+λV(x)u=K(x)|u|p−2u     x∈ℝN,     u→0 as |x| →+∞, where N ≥ 3, λ > 0 is a parameter, 2 >p > 2NN−2, and the potentials V(x) and K(x) satisfy some suitable conditions. By using the method based on invariant sets of the descending flow, we obtain the existence of a positive ground state solution and a ground state sign-changing solution of the above equation for small λ, which is a complement of the results obtained by Wang and Zhou in [J. Math. Phys. 52, 113704, 2011].