Multiple solutions for the nonlinear Schrödinger-Poisson system with a partial confinement

Abstract

We study the following nonlinear Schrödinger-Poisson system with a partial confinement$\{\begin{array}{cc}-\Delta u+(x_1^2+x_2^2)u+\lambda \varphi \left(x\right)u=|u{|}^{p-2}u,& x\in R^3,\\ -\Delta \varphi =u^2,& x\in R^3,\end{array}$ where $p\in (2,6)$ and $\lambda >0$ is a parameter. The existence and nonexistence results are established by variational methods, depending on the parameters p and λ. It turns out that $p=3$ is a critical value for the existence of solutions.

Our results can be viewed as an extension of the results of Ruiz [33] concerning the nonlinear Schrödinger-Poisson equation with a positive constant potential. However, due to the presence of partial confinement, the Nehari-Pohozaev manifold method is no longer applicable in this paper for $p\in (3,\frac{16}{5}]$. We need to explore the more complicated underlying functional geometry with a different variational approach. Moreover, we also construct saddle type nodal solutions whose nodal domains meet at the origin.