Normalized solutions for a critical fractional Choquard equation with a nonlocal perturbation

Abstract

In this article, we study the fractional critical Choquard equation with a nonlocal perturbation: (−Δ)su=λu+α(Iμ*∣u∣q)∣u∣q−2u+(Iμ*∣u∣2μ,s*)∣u∣2μ,s*−2u,inRN,{\left(-{\Delta })}^{s}u=\lambda u+\alpha \left({I}_{{\mu }^{* }}\hspace{-0.25em}{| u| }^{q}){| u| }^{q-2}u+\left({I}_{{\mu }^{* }}\hspace{-0.25em}{| u| }^{{2}_{\mu ,s}^{* }}){| u| }^{{2}_{\mu ,s}^{* }-2}u,\hspace{1em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{N}, having prescribed mass ∫RNu2dx=c2,\mathop{\int }\limits_{{{\mathbb{R}}}^{N}}{u}^{2}{\rm{d}}x={c}^{2}, where s∈(0,1),N>2s,00,c>0s\in \left(0,1),N\gt 2s,0\lt \mu \lt N,\alpha \gt 0,c\gt 0, and Iμ(x){I}_{\mu }\left(x) is the Riesz potential given by Iμ(x)=Aμ∣x∣μwithAμ=Γμ22N−μπN⁄2ΓN−μ2,{I}_{\mu }\left(x)=\frac{{A}_{\mu }}{{| x| }^{\mu }}\hspace{1em}\hspace{0.1em}\text{with}\hspace{0.1em}\hspace{0.33em}{A}_{\mu }=\frac{\Gamma \left(\phantom{\rule[-0.75em]{}{0ex}},\frac{\mu }{2}\right)}{{2}^{N-\mu }{\pi }^{N/2}\Gamma \left(\phantom{\rule[-0.75em]{}{0ex}},\frac{N-\mu }{2}\right)}, and 2N−μN<q<2μ,s*=2N−μN−2s\frac{2N-\mu }{N}\lt q\lt {2}_{\mu ,s}^{* }=\frac{2N-\mu }{N-2s} is the fractional Hardy-Littlewood-Sobolev critical exponent. Under the L2{L}^{2}-subcritical perturbation α(Iμ*∣u∣q)∣u∣q−2u\alpha \left({I}_{{\mu }^{* }}\hspace{-0.25em}{| u| }^{q}){| u| }^{q-2}u with exponent 2N−μN<q<2N−μ+2sN\frac{2N-\mu }{N}\lt q\lt \frac{2N-\mu +2s}{N}, we obtain the existence of normalized ground states and mountain-pass-type solutions. Meanwhile, for the L2{L}^{2}-critical and L2{L}^{2}-supercritical cases 2N−μ+2sN≤q<2N−μN−2s\frac{2N-\mu +2s}{N}\le q\lt \frac{2N-\mu }{N-2s}, we also prove that the equation has ground states of mountain-pass-type.

Keywords

• fractional choquard equation
• normalized solutions
• critical exponent
• variational methods
• 35j62
• 35j50
• 35b65