Multiple and least energy sign-changing solutions for Schr¨odinger-Poisson equations in R3 with restraint

In this paper, we study the existence of multiple sign-changing solutions with a prescribed Lp+1−norm and the existence of least energy sign-changing restrained solutions for the following nonlinear Schr¨odinger-Poisson system:  −△u + u + ϕ(x)u = λ|u|p−1u, in R3, −△ϕ(x) = |u|2, in R3. By choosing a proper functional restricted on some appropriate subset to using a method of invariant sets of descending flow, we prove that this system has infinitely many sign-changing solutions with the prescribed Lp+1−norm and has a least energy for such sign-changing restrained solution for p ∈ (3, 5). Few existence results of multiple sign-changing restrained solutions are available in the literature. Our work generalize some results in literature.