Electron. J. Differential Equations, Vol. 2019 (2019), No. 102, pp. 1-16.

Non-trivial solutions of fractional Schrodinger-Poisson systems with sum of periodic and vanishing potentials

Mingzhu Yu, Haibo Chen

Abstract:
We consider the fractional Schrodinger-Poisson system

where $\alpha,\beta\in(0,1]$, $4\alpha+2\beta>3$, $4\leq q<2_{\alpha}^{\ast}$, K(x), $\Gamma(x)$ and f(x,u) are periodic in x, V is coercive or $V=V_{\rm per}+V_{\rm loc}$ is a sum of a periodic potential $V_{\rm per}$ and a localized potential $V_{\rm loc}$. If f has the subcritical growth, but higher than $\Gamma(x)|u|^{q-2}u$, we establish the existence and nonexistence of ground state solutions are dependent on the sign of $V_{\rm loc}$. Moreover, we prove that such a problem admits infinitely many pairs of geometrically distinct solutions provided that V is periodic and f is odd in u. Finally, we investigate the existence of ground state solutions in the case of coercive potential V.

Submitted March 22, 2018. Published September 4, 2019.
Math Subject Classifications: 35B38, 35G99.
Key Words: Fractional Schrodinger-Poisson system; coercive potential; periodic and localized potential; Nehari manifold; variational method.

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Mingzhu Yu
School of Mathematics and Statistics
Central South University
Changsha, 410083 Hunan, China
email: yumz_math@csu.edu.cn
Haibo Chen
School of Mathematics and Statistics
Central South University
Changsha, 410083 Hunan, China
email: math_chb@163.com

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