In this work, we study the existence of a weak solution to the following quasi linear elliptic problem involving the fractional p-Laplacian operator, a Hardy potential, and multiple critical Sobolev nonlinearities with singularities,$$(-{\Delta}_{p})^{s}u - {\mu}\frac{|u|^{p-2}u}{|x|^{ps}}=\frac{|u|^{p^{*}_{s}(\beta)-2_{u}}}{|x|^{\beta}}+\frac{|u|^{p^{*}_{s}{(\alpha)-2_u}}}{|x|^{\alpha}},$$where \(x \in \mathbb{R}^{N}, u \in D^{s,p}(\mathbb{R}^{N}), 0 < s < 1,1 < p < +\infty, N > sp,\)\(0 < \alpha < sp, 0< \beta < sp, \beta \neq \alpha, \mu < \mu_{H} := {\rm inf}_{u{\in}D^{s,p}(\mathbb{R}^{N})\backslash{\{0}\}}{[u]^{p}_{s,p}/\|u\|^{p}_{s,p}> 0.} \) To prove the existence of solution to the problem, we have to formulate a refined version of the concentration-compactness principle and, as an independent result, we have to show that the extremals for the Sobolev inequality are attained.

中文翻译：

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具有多个临界Hardy-Sobolev非线性的分数p-Laplacian问题

在这项工作中，我们研究了以下分数线性椭圆问题的弱解决方案，该问题涉及分数p -Laplacian算子，Hardy势以及具有奇异性$$（-{\ Delta} _ {p }）^ {s} u-{\ mu} \ frac {| u | ^ {p-2} u} {| x | ^ {ps}} = \ frac {| u | ^ {p ^ {*} _ {s}（\ beta）-2_ {u}}} {| x | ^ {\ beta}} + \ frac {| u | ^ {p ^ {*} _ {s} {{\ alpha）-2_u} }} {| x | ^ {\ alpha}}，$$其中\（x \ in \ mathbb {R} ^ {N}，u \ in D ^ {s，p}（\ mathbb {R} ^ {N }），0 <s <1,1 <p <+ \ infty，N> sp，\）\（0 <\ alpha <sp，0 <\ beta <sp，\ beta \ neq \ alpha，\ mu <\ mu_ {H}：= {\ rm inf} _ {u {\ in} D ^ {s，p}（\ mathbb {R} ^ {N}）\反斜杠{\ {0} \}} {[u] ^ {p} _ {s，p} / \ | u \ | ^ {p} _ {s，p}>0。} \） 为了证明问题的解决方案的存在，我们必须制定浓度紧致原理的精简版本，作为独立的结果，我们必须证明获得了Sobolev不等式的极值。