Multiple solutions for Schrödinger-Poisson systems with critical nonlocal term

Zuji Guo

DOI: http://dx.doi.org/10.12775/TMNA.2019.077

Abstract


This paper is concerned with the existence of positive bound state solutions for Schrödinger-Poisson systems with critical nonlocal term: \begin{equation*} \begin{cases} -\Delta u=\phi|u|^{3}u+\lambda Q(x)|u|^{q-2}u &\text{in } \mathbb{R}^3, \\ -\Delta \phi=|u|^5 & \text{in } \mathbb{R}^3. \end{cases} \leqno(\mathcal{P}) \end{equation*} Under certain assumptions on $Q$ and $\lambda$, we prove that $(\mathcal{P})$ has multiple positive bound state solutions by decomposition the Nehari manifold and fine estimates.

Keywords


Critical nonlocal term; ground state solution; Nehari manifold; Schrödinger-Poisson systems

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