Sign-changing solutions for the nonlinear Schrödinger–Poisson system in $$\mathbb {R}^3$$ R 3

Abstract

We are interested in the existence and energy estimate of sign-changing solutions for a Schrodinger–Poisson problem on $$\mathbb {R}^3$$ . Because the so-called nonlocal term $$\lambda \phi _u (x) u$$ is involving in the equation, the variational functional of the equation has totally different properties from the case of $$\lambda =0$$ . By introducing some new ideas, we prove, via a constraint variational method combining the Brouwer degree theory, that the problem has a sign-changing solution under suitable conditions. When $$\lambda =0$$ , it is known that any sign-changing solution of the problem has an energy exceeding twice the least energy. However, it is still unknown what would happen when $$\lambda >0$$ . We partially answer this question for $$\lambda >0$$ being small.