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

Abstract

We study the following fractional Kirchhoff-Choquard equation: a+b∫RN(−Δ)s2u2dx(−Δ)su+V(x)u=(Iμ*F(u))f(u),x∈RN,u∈Hs(RN),\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,ba,b are positive constants, N>2sN\gt 2s, μ∈((N−4s)+,N)\mu \in \left({\left(N-4s)}_{+},N), s∈(0,1)s\in \left(0,1), and Iμ{I}_{\mu } is the Riesz potential. Considering the case that nonlinearity ff 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.

Keywords

• kirchhoff-choquard equation
• fractional
• ground-state solution
• pohozaev identity
• 35r11
• 49j35