On the regularity of positive solutions of a class of Choquard type equations

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This paper is concerned with positive solutions of a class of Choquard type equations. Such equations are equivalent to integral systems involving the Bessel potential and the Riesz potential. By using two regularity lifting lemmas introduced by Chen and Li [2], we study the regularity for integrable solutions u. We first use the Hardy–Littlewood–Sobolev inequality to obtain an integrability result. Then, it is improved to \({u \in L^s(R^n)}\) for all \({s \in [1, \infty]}\) by an iteration. Next, we use the properties of the contraction map and the shrinking map to prove that u is Lipschitz continuous. Finally, we establish the smoothness of u by a bootstrap argument. Our technique can also be used to handle other integral systems involving the Riesz potential or the Bessel potential, such as the Hartree type equations.