Journal of Pseudo-Differential Operators and Applications ( IF 0.9 ) Pub Date : 2021-02-19 , DOI: 10.1007/s11868-021-00382-2 Akasmika Panda , Debajyoti Choudhuri , Kamel Saoudi
This article concerns about the existence of a positive SOLA (Solutions Obtained as Limits of Approximations) for the following singular critical Choquard problem involving fractional power of Laplacian and a critical Hardy potential.
$$\begin{aligned} \begin{aligned} (-\Delta )^su-\alpha \frac{u}{|x|^{2s}}&=\lambda u+ u^{-\gamma }+\beta \left( \int _{\Omega }\frac{u^{2_b^*}(y)}{|x-y|^b}dy\right) u^{2_b^*-1}+\mu ~\text {in}~\Omega ,\\ u&>0~\text {in}~\Omega ,\\ u&= 0~\text {in}~\mathbb {R}^N{\setminus }\Omega . \end{aligned} \end{aligned}$$(0.1)Here, \(\Omega \) is a bounded domain of \(\mathbb {R}^N\), \(s\in (0,1)\), \(\alpha ,\lambda \) and \(\beta \) are positive real parameters, \(N>2s\), \(\gamma \in (0,1)\), \(0<b<\min \{N,4s\}\), \(2_b^*=\frac{2N-b}{N-2s}\) is the critical exponent in the sense of Hardy–Littlewood–Sobolev inequality and \(\mu \) is a bounded Radon measure in \(\Omega \).
中文翻译：
包含奇异非线性和ra测量的临界分数次方差问题
本文关注的是以下奇异的临界Choquard问题（涉及拉普拉斯分数功率和临界Hardy势）的正SOLA（作为近似极限获得的解决方案）的存在。
$$ \ begin {aligned} \ begin {aligned（（\\ Delta）^ su- \ alpha \ frac {u} {| x | ^ {2s}}＆= \ lambda u + u ^ {-\ gamma} + \ beta \ left（\ int _ {\ Omega} \ frac {u ^ {2_b ^ *}（y）} {| xy | ^ b} dy \ right）u ^ {2_b ^ *-1} + \ mu〜\文本{in}〜\ Omega，\\ u＆> 0〜\文本{in}〜\ Omega，\\ u＆= 0〜\文本{in}〜\ mathbb {R} ^ N {\ setminus} \ Omega。\ end {aligned} \ end {aligned} $$（0.1）在这里，\（\ Omega \）是\（\ mathbb {R} ^ N \），\（s \ in（0,1）\），\（\ alpha，\ lambda \）和\（ \ beta \）是正实数参数\（N> 2s \），\（\ gamma \ in（0,1）\），\（0 <b <\ min \ {N，4s \} \），\ （2_b ^ * = \ frac {2N-b} {N-2s} \）是Hardy–Littlewood–Sobolev不等式意义上的关键指数，\（\ mu \）是\（\ Omega中的有界Radon量度\）。
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