Ground-state solutions for fractional Kirchhoff-Choquard equations with critical growth

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Abstract We study the following fractional Kirchhoff-Choquard equation: a + b ∫ R N ( − Δ ) s 2 u 2 d x ( − Δ ) s u + V ( x ) u = ( I μ * F ( u ) ) f ( u ) , x ∈ R N , u ∈ H s ( R N ) , \left\{\begin{array}{l}\left(a+b\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}{\left|{\left(-\Delta )}^{\frac{s}{2}}u\right|}^{2}{\rm{d}}x\right){\left(-\Delta )}^{s}u+V\left(x)u=\left({I}_{\mu }* F\left(u))f\left(u),\hspace{1em}x\in {{\mathbb{R}}}^{N},\hspace{1.0em}\\ u\in {H}^{s}\left({{\mathbb{R}}}^{N}),\hspace{1.0em}\end{array}\right. where a , b a,b are positive constants, N > 2 s N\gt 2s , μ ∈ ( ( N − 4 s ) + , N ) \mu \in \left({\left(N-4s)}_{+},N) , s ∈ ( 0 , 1 ) s\in \left(0,1) , and I μ {I}_{\mu } is the Riesz potential. Considering the case that nonlinearity f f has critical growth, combining a monotonicity trick and global compactness lemma, we prove that the equation has a ground-state solution. Moreover, we study the regularity of ground-state solutions to the above equation, which extends the results in Moroz-Van Schaftingen [Existence of groundstates for a class of nonlinear Choquard equations, Trans. Amer. Math. Soc. 367 (2015), 6557–6579] to the fractional Laplacian case.