This paper concerns with the existence and regularity of solutions for the following Choquard type equation,$$-\Delta_u = \big(I_{\mu} * F(u)\big) f(u) {\rm in} \mathbb{R}^3, \quad \quad (P)$$where \({I_\mu = \frac{1}{|x|^\mu}, 0 < \mu < 3}\), is the Riesz potential, \({F(s)}\) is the primitive of the continuous function f(s), and \({I_{\mu} * F(u)}\) denotes the convolution of \({I_{\mu}}\) and F(u). By using the variational method, we prove that problem (P), in the zero mass case, possesses at least a nontrivial solution under certain conditions on f.

中文翻译：

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一类零质量Chquard方程解的存在性和正则性

本文关注以下Choquard型方程的解的存在性和规则性，$$-\ Delta_u = \ big（I _ {\ mu} * F（u）\ big）f（u）{\ rm in} \ mathbb {R} ^ 3，\ quad \ quad（P）$$其中\（{I_ \ mu = \ frac {1} {| x | ^ \ mu}，0 <\ mu <3} \）是里兹势\（{F（s）} \）是连续函数f（s）的本原，\（{I _ {\ mu} * F（u）} \）表示\（{I_ { \ mu}} \）和F（u）。通过使用变分方法，我们证明了问题（P）在零质量情况下在f上的某些条件下至少具有非平凡解。