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

Abstract

Abstract In this article, we study the fractional critical Choquard equation with a nonlocal perturbation: ( − Δ ) s u = λ u + α ( I μ * ∣ u ∣ q ) ∣ u ∣ q − 2 u + ( I μ * ∣ u ∣ 2 μ , s * ) ∣ u ∣ 2 μ , s * − 2 u , in R N , {\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 ∫ R N u 2 d x = c 2 , \mathop{\int }\limits_{{{\mathbb{R}}}^{N}}{u}^{2}{\rm{d}}x={c}^{2}, where s ∈ ( 0 , 1 ) , N > 2 s , 0 < μ < N , α > 0 , c > 0 s\in \left(0,1),N\gt 2s,0\lt \mu \lt N,\alpha \gt 0,c\gt 0 , and I μ ( x ) {I}_{\mu }\left(x) is the Riesz potential given by I μ ( x ) = A μ ∣ x ∣ μ with A μ = Γ μ 2 2 N − μ π N ⁄ 2 Γ N − μ 2 , {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 2 N − μ N < q < 2 μ , s * = 2 N − μ N − 2 s \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 L 2 {L}^{2} -subcritical perturbation α ( I μ * ∣ u ∣ q ) ∣ u ∣ q − 2 u \alpha \left({I}_{{\mu }^{* }}\hspace{-0.25em}{| u| }^{q}){| u| }^{q-2}u with exponent 2 N − μ N < q < 2 N − μ + 2 s N \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 L 2 {L}^{2} -critical and L 2 {L}^{2} -supercritical cases 2 N − μ + 2 s N ≤ q < 2 N − μ N − 2 s \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.