manuscripta mathematica ( IF 0.5 ) Pub Date : 2021-05-12 , DOI: 10.1007/s00229-021-01309-3 Alessio Fiscella , Pawan Kumar Mishra
The paper deals with the following singular fractional problem
$$\begin{aligned} \left\{ \begin{array}{lll} M\left( \displaystyle \iint _{{\mathbb {R}}^{2N}}\frac{|u(x)-u(y)|^2}{|x-y|^{N+2s}}dxdy\right) (-\Delta )^{s} u-\mu \displaystyle \frac{u}{|x|^{2s}}= \lambda f(x)u^{-\gamma }+ g(x){u^{2^*_s-1}}&{}\;\; \text {in}\; \Omega ,\\ u>0&{} \;\; \text {in}\; \Omega ,\\ u=0&{}\;\;\text {in}\;{\mathbb {R}}^N\setminus \Omega , \end{array}\right. \end{aligned}$$where \(\Omega \subset {\mathbb {R}}^N\) is an open bounded domain, with \(0\in \Omega \), dimension \(N>2s\) with \(s\in (0,1)\), \(2^*_s=2N/(N-2s)\) is the fractional critical Sobolev exponent, \(\lambda \) and \(\mu \) are positive parameters, exponent \(\gamma \in (0,1)\), M models a Kirchhoff coefficient, f is a positive weight while g is a sign-changing function. The main feature and novelty of our problem is the combination of the critical Hardy and Sobolev nonlinearities with the bi-nonlocal framework and a singular nondifferentiable term. By exploiting the Nehari manifold approach, we provide the existence of at least two positive solutions.
中文翻译：
具有奇异和临界Sobolev非线性的分数Kirchhoff Hardy问题
本文处理以下奇异分数问题
$$ \ begin {aligned} \ left \ {\ begin {array} {lll} M \ left（\ displaystyle \ iint _ {{\ mathbb {R}} ^ {2N}} \ frac {| u（x）- u（y）| ^ 2} {| xy | ^ {N + 2s}} dxdy \ right）（-\ Delta）^ {s} u- \ mu \ displaystyle \ frac {u} {| x | ^ {2s }} = \ lambda f（x）u ^ {-\ gamma} + g（x）{u ^ {2 ^ * _ s-1}}＆{} \; \; \; \ text {in} \; \ Omega，\\ u> 0＆{} \; \; \ text {in} \; \ Omega，\\ u = 0＆{} \; \; \ text {in} \; {\ mathbb {R}} ^ N \ setminus \ Omega，\ end {array} \ right。\ end {aligned} $$其中\（\ Omega \ subset {\ mathbb {R}} ^ N \）是一个开放边界域，其中\（0 \ in \ Omega \），尺寸\（N> 2s \）具有\（s \ in（ 0,1）\），\（2 ^ * _ s = 2N /（N-2s）\）是分数临界Sobolev指数，\（\ lambda \）和\（\ mu \）是正参数，指数\（ \ gamma \ in（0,1）\），M模拟一个基尔霍夫系数，f是一个正权重，而g是一个符号转换功能。我们问题的主要特征和新颖性是临界Hardy和Sobolev非线性与双非局部框架和奇异不可微项的组合。通过利用Nehari流形方法，我们提供了至少两个正解的存在。
京公网安备 11010802027423号