| Contents |
| The Choquard equation, also known as Hartree equation or nonlinear Schr\"odinger-Newton equation is a stationary nonlinear Schr\"odinger type equation where the nonlinearity is coupled with a nonlocal convolution term associated with an attractive gravitational potential. We present a survey of recent results on Choquard type equations, focusing on Liouville type nonexistence theorems and sharp a priori decay estimates of the solutions and super-solutions. The techniques of proofs include an integral form of Phragmen-Lindel\"of type comparison estimates and a nonlocal nonlinear extension of the Agmon-Allegretto-Piepenbrink positivity principle which relates the existence of a positive super-solutions to an integral inequality. |
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