Multiple positive bound states for critical Schr\"odinger-Poisson systems

Giovanna Cerami; Riccardo Molle

Multiple positive bound states for critical Schr\"odinger-Poisson systems

Analysis of PDEs 2018-02-08 v1

Authors: Giovanna Cerami , Riccardo Molle

Abstract

Using variational methods we prove some results about existence and multiplicity of positive bound states of to the following Schr\"odinger-Poisson system: $\left\{ \begin{array}{l} \vspace{2mm} -\Delta u+V(x)u+K(x)\phi(x)u=u^5 -\Delta \phi =K(x)u^2\qquad x\in\R^3 \end{array}\right.\quad\quad (SP)$ We remark that $(SP)$ exhibits a "double" lack of compactness because of the unboundedness of $\R^3$ and the critical growth of the nonlinear term and that in our assumptions ground state solutions of $(SP)$ do not exist.

Keywords

nonlinear schrodinger equations ground states nonlinear schrödinger equations

Cite

@article{arxiv.1802.02539,
  title  = {Multiple positive bound states for critical Schr\"odinger-Poisson systems},
  author = {Giovanna Cerami and Riccardo Molle},
  journal= {arXiv preprint arXiv:1802.02539},
  year   = {2018}
}

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