Ground state solutions for planar Schrödinger-Poisson system involving subcritical and critical exponential growth with convolution nonlinearity

Ground states for nonlinear fractional Schrödinger–Poisson systems with general convolution nonlinearities

“…has been widely studied in recent years; see earlier studies [1][2][3][4] and references therein. When 𝜙(x) = 0, system (1.2) reduces to the Choquard equation…”

“…When $$$ s=1 $$$, the Schrödinger–Poisson type problem $$$$ \left\{\begin{array}{ll}-\Delta u+V(x)u+\phi u=\left({I}_{\mu}\ast f(u)\right)F(u),\kern0.30em & \kern0.5em \mathrm{in}\kern0.5em {\mathrm{\mathbb{R}}}^3,\\ {}-\Delta \phi ={u}^2,& \kern0.5em \mathrm{in}\kern0.5em {\mathrm{\mathbb{R}}}^3\end{array}\right. $$$$ has been widely studied in recent years; see earlier studies [1–4] and references therein. When $$$ \phi (x)=0 $$$, system () reduces to the Choquard equation $$$$ -\Delta u+V(x)u=\left({I}_{\mu}\ast f(u)\right)F(u),\kern0.40em u\in {H}^1\left({\mathrm{\mathbb{R}}}^3\right).$$…”

“…Assume that (V 2 ), (𝑓 1 ), and (𝑓 ′ 4 ) hold. Then for all u ∈ H s (R 3 ) and t ≥ 0, Φ(u) ≥ Φ(𝜃u) + 1 − 𝜃 4 4…”

Ground state solutions for fractional Choquard-Schrödinger-Poisson system with critical growth

“…has been widely studied in recent years, where V ∈ C (R 3 , [0, +∞)) and f ∈ C (R, R); see [3,4,5] and the references therein. When φ (x) = 0, system (1.2) reduces to the Choquard equation…”