Geodesic
FRACTIONAL SCHRÖDINGER–POISSON SYSTEM WITH SINGULARITY: EXISTENCE, UNIQUENESS, AND ASYMPTOTIC BEHAVIOR
Glasgow mathematical journal, Tome 63 (2021) no. 1, pp. 179-192

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In this paper, we consider the following fractional Schrödinger–Poisson system with singularity \begin{equation*}\left \{\begin{array}{lcl} ({-}\Delta)^s u+V(x)u+\lambda \phi u = f(x)u^{-\gamma}, &&\quad x\in\mathbb{R}^3,\\ ({-}\Delta)^t \phi = u^2, &&\quad x\in\mathbb{R}^3,\\ u>0,&&\quad x\in\mathbb{R}^3,\end{array}\right.\end{equation*}where 0 < γ < 1, λ > 0 and 0 < s ≤ t < 1 with 4s + 2t > 3. Under certain assumptions on V and f, we show the existence, uniqueness, and monotonicity of positive solution uλ using the variational method. We also give a convergence property of uλ as λ → 0, when λ is regarded as a positive parameter. 
YU, SHENGBIN; CHEN, JIANQING. FRACTIONAL SCHRÖDINGER–POISSON SYSTEM WITH SINGULARITY: EXISTENCE, UNIQUENESS, AND ASYMPTOTIC BEHAVIOR. Glasgow mathematical journal, Tome 63 (2021) no. 1, pp. 179-192. doi: 10.1017/S0017089520000099
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     author = {YU, SHENGBIN and CHEN, JIANQING},
     title = {FRACTIONAL {SCHR\"ODINGER{\textendash}POISSON} {SYSTEM} {WITH} {SINGULARITY:} {EXISTENCE,} {UNIQUENESS,} {AND} {ASYMPTOTIC} {BEHAVIOR}},
     journal = {Glasgow mathematical journal},
     pages = {179--192},
     year = {2021},
     volume = {63},
     number = {1},
     doi = {10.1017/S0017089520000099},
     url = {http://geodesic.mathdoc.fr/articles/10.1017/S0017089520000099/}
}
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