Positive solutions for a nonlocal Schrödinger–Newton system involving critical nonlinearity

Existence of Solutions for a Schrödinger–Poisson System with Critical Nonlocal Term and General Nonlinearity

“…But there are currently only a few results for the following Schrödinger-Poisson systems with critical nonlocal terms in a bounded domain [11][12][13]:…”

“…Later, when q = 1, η = 1, gðx, uÞ = λu, where λ is a real number, the authors in [12] studied system (1); they proved existence and nonexistence results of positive solutions when N = 3 and existence of solutions in both the resonance and the nonresonance case for higher dimensions. For the case q = 1, gðx, uÞ = λf ðxÞ, λ > 0 is a real number, f ≥ 0, and f ∈ L 2 * /ð2 * −1Þ ðℝ N Þ, in [13]; when η = 1, authors proved that system (1) has at least two positive solutions if 0 < λ < λ * for some λ * >0 small enough, and when η = −1, system (1) has at least one positive solution for any λ > 0.…”