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Small linear perturbations of fractional Choquard equations with critical exponent
He, Xiaoming ; Rǎdulescu, Vicenţiu, 1958-
We are concerned with the qualitative analysis of positive solutions to the fractional Choquard equation ▫$$ \begin{cases} (-\Delta)^s u + a(x)u = (I_\alpha \ast |u|^{2^\ast_{\alpha, s}}) ...|u|^{2^\ast_{\alpha, s}-2}u, & x\in \mathbb{R}^N \\ u \in D^{s,2} (\mathbb{R}^N), \; u(x)>0, & x\in \mathbb{R}^N, \end{cases}$$▫ where ▫$I_\alpha(x)$▫ is the Riesz potential, ▫$s \in (0, 1)$▫, ▫$N>2s$▫, ▫$0<\alpha<\min \{N, 4s\}$▫, and ▫$2^\ast_{\alpha, s} = \frac{2N-\alpha}{N-2s}$▫ is the fractional critical Hardy-Littlewood-Sobolev exponent. We first establish a nonlocal global compactness property in the framework of fractional Choquard equations. In the second part of this paper, we prove that the equation has at least one positive solution in the case of small perturbations of the potential that describes the linear term.
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