Regularity for critical fractional Choquard equation with singular potential and its applications

Abstract

Abstract We study the following fractional Choquard equation <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="block"> <m:msup> <m:mrow> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mo>−</m:mo> <m:mi>Δ</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:mrow> <m:mrow> <m:mi>s</m:mi> </m:mrow> </m:msup> <m:mi>u</m:mi> <m:mo>+</m:mo> <m:mfrac> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:msup> <m:mrow> <m:mo>∣</m:mo> <m:mi>x</m:mi> <m:mo>∣</m:mo> </m:mrow> <m:mrow> <m:mi>θ</m:mi> </m:mrow> </m:msup> </m:mrow> </m:mfrac> <m:mo>=</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:msub> <m:mrow> <m:mi>I</m:mi> </m:mrow> <m:mrow> <m:mi>α</m:mi> </m:mrow> </m:msub> <m:mo>*</m:mo> <m:mi>F</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mi>f</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>,</m:mo> <m:mspace width="1em"/> <m:mi>x</m:mi> <m:mo>∈</m:mo> <m:msup> <m:mrow> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> <m:mrow> <m:mi>N</m:mi> </m:mrow> </m:msup> <m:mo>,</m:mo> </m:math> {\left(-\Delta )}^{s}u+\frac{u}{{| x| }^{\theta }}=({I}_{\alpha }* F\left(u))f\left(u),\hspace{1em}x\in {{\mathbb{R}}}^{N}, where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>N</m:mi> <m:mo>⩾</m:mo> <m:mn>3</m:mn> </m:math> N\geqslant 3 , <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>s</m:mi> <m:mo>∈</m:mo> <m:mfenced open="(" close=")"> <m:mspace depth="0.75em"/> <m:mrow> <m:mfrac> <m:mrow> <m:mn>1</m:mn> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:mfrac> <m:mo>,</m:mo> <m:mn>1</m:mn> </m:mrow> </m:mfenced> </m:math> s\in \left(\phantom{\rule[-0.75em]{}{0ex}},\frac{1}{2},1\right) , <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>α</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mn>0</m:mn> <m:mo>,</m:mo> <m:mi>N</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> \alpha \in \left(0,N) , <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>θ</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mn>0</m:mn> <m:mo>,</m:mo> <m:mn>2</m:mn> <m:mi>s</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> \theta \in \left(0,2s) , and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mi>I</m:mi> </m:mrow> <m:mrow> <m:mi>α</m:mi> </m:mrow> </m:msub> </m:math> {I}_{\alpha } is the Riesz potential. The main purpose of this article is twofold. We first study the regularity of weak solutions for the aforementioned equation with critical nonlinearity, which extends the results of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>θ</m:mi> <m:mo>=</m:mo> <m:mn>0</m:mn> </m:math> \theta =0 in Moroz-Van Schaftingen [ Existence of groundstates for a class of nonlinear Choquardequations , Trans. Amer. Math. Soc. 367 (2015), no. 9, 6557–6579]. Then, as an application of the regularity results, we establish the existence of ground state so

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