Abstract:

A class of nonlinear Schrödinger-Poisson equations is considered,

$ -\varDelta u+\left( V\left( x \right) -\frac{\mu}{\left| x \right|} \right) u+\lambda \left( \frac{1}{\left| x \right|}*\left| u \right|^2 \right) u=\left| u \right|^{p-1}u,\ x\in \mathbb{R}^3, \ \lambda>0, $

where $V(x)$ denotes a periodic potential function, $\mu,\ \lambda \in \mathbb{R}$ are parameters, and $p\in \left( 1,2 \right)$. By employing variational methods and the Cerami sequence profile decomposition, the existence and nonexistence of ground state solutions to the above equation are established under suitable assumptions on the parameters.

Key words: ground state, variational methods, Nehari manifold, periodic potential, Schrödinger-Poisson equation