We consider two fractional in time nonlinear Sobolev-type inequalities involving potential terms, where the fractional derivatives are defined in the sense of Caputo. For both problems, we study the existence and nonexistence of nontrivial local weak solutions. Namely, we show that there exists a critical exponent according to which we have existence or nonexistence.

中文翻译：

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时间分数阶SOBOLEV型不等式的局部弱解

我们考虑两个涉及时间项的时间分数阶非线性Sobolev型不等式，其中分数导数是在Caputo的意义上定义的。对于这两个问题，我们研究非平凡局部弱解的存在和不存在。即，我们表明存在一个关键指数，据此我们可以存在或不存在。