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Singular doubly nonlocal elliptic problems with Choquard type critical growth nonlinearities. (English) Zbl 1477.49022

The paper studies the existence and global multiplicity of positive weak solutions to a new doubly nonlocal singular problem \[\left\{ \begin{array}{l} (-\Delta )^s u=u^{-q}+\lambda \Big(\int_\Omega \frac{\vert u \vert^{2_\mu^*}(y)}{\vert x-y \vert^\mu}dy\Big)\vert u \vert^{2_\mu^*-2}u,\;\;\; u>0\;\;\text{ in \;} \Omega,\\ u=0 \text{ in }\mathbb{R}^N\setminus \Omega, \end{array}\right. \] where \(q>0, N>2s, s\in (0,1), 2_\mu^*=\frac{2N-\mu}{N-2s}\), \(\Omega\) is a bounded domain in \(\mathbb{R}^N\) with smooth boundary and \((-\Delta )^s\) is the fractional Laplacian defined by \[ (-\Delta )^s u(x)=-\mathrm{P. V.} \int_{\mathbb{R}^N}\frac{u(x)-u(y)}{\vert x-y\vert^{N+2s}}dy, \] where \(-\mathrm{P. V.} \) is the Cauchy principal value. A weak comparison principle and the optimal Sobolev regularity are also provided.

MSC:

49J52 Nonsmooth analysis
35A15 Variational methods applied to PDEs
35S15 Boundary value problems for PDEs with pseudodifferential operators
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
35D30 Weak solutions to PDEs
35B09 Positive solutions to PDEs
35R11 Fractional partial differential equations
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