Radial ground state sign-changing solutions for asymptotically cubic or super-cubic fractional Schrödinger-Poisson systems

ABSTRACT

In this paper, we consider the following fractional Schrödinger-Poisson system: $\begin{array}{ll}(-\mathrm{△}{)}^{s}u+V(|x|)u+\lambda \phi (x)u=f(|x|,u),& x\in {\mathbb{R}}^3,\\ (-\mathrm{△}{)}^t\phi =u^2,& x\in {\mathbb{R}}^3,\end{array}$ where $s,t\in (0,1],4s+2t>3$, λ is a positive parameter and f satisfies asymptotically cubic or super-cubic growth that is very different from super-cubic growth in previous literatures. Without assuming the usual Nehari-type monotonic condition on $f\left(\tau \right)/|\tau {|}^3$, we establish the existence of one radial ground state sign-changing solution $u_{\lambda }$ with precisely two nodal domains. Furthermore, we also prove that the energy of any radial sign changing solution is strictly larger than two times the least energy and give a convergence property of $u_{\lambda }$ as $\lambda \searrow 0$. Our results unify both asymptotically cubic and super-cubic nonlinearities, which are new even for s=t=1.

COMMUNICATED BY:

• D. Mitrea

KEYWORDS:

• Fractional Schrödinger-Poisson sysytem
• ground state sign-changing solution
• asymptotically cubic and super-cubic growth

AMS SUBJECT CLASSIFICATIONS:

• 35J20
• 35J25
• 35J60

Acknowledgments

The authors are grateful to the anonymous referees for their valuable suggestions and comments.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work is partially supported by the Fundamental Research Funds for the Central Universities of Central South University (No: 502211712) and the National Natural Science Foundation of China (No: 11571370).