Fractional Schrödinger-Poisson-Slater system in one dimension

Abstract

In this paper we study local and global well-posedness of the following Cauchy problem: $$\{\begin{array}{rl}i{\partial }_t\Psi +\frac{1}{2}{\Delta }_x\Psi =A_0\Psi +\alpha |\Psi {|}^{\gamma -1}\Psi ,& (t,x)\in \mathbb{R}\times \mathbb{R},\\ (-{\Delta }_x{)}^{\sigma /2}{A}_0=|\Psi {|}^2,& \\ \Psi (0,\cdot )=f,& \end{array}$$ with $\sigma \in (0,1)$, $\alpha =\pm 1$, $1<\gamma \le 5$, in the spaces $L^2(\mathbf{R})$ and $H^1(\mathbf{R})$.

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• Fractional Schrödinger-Poisson systems with a general subcritical or critical nonlinearity
• Existence and multiplicity results for the fractional Schrodinger-Poisson systems
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