Multiplicity of solutions for a class of fractional Choquard–Kirchhoff equations involving critical nonlinearity

Abstract

The aim of this paper is to investigate the multiplicity of solutions to the following nonlocal fractional Choquard–Kirchhoff type equation involving critical exponent,

$$\begin{aligned}&\left( a+b[u]_{s,p}^p\right) (-\Delta )_p^su=\int _{\mathbb {R}^N}\frac{|u(y)|^{p_{\mu ,s}^*}}{|x-y|^{\mu }}dy|u|^{p_{\mu ,s}^*-2}u +\lambda h(x)|u|^{q-2}u\quad&\text{ in } \,\,\mathbb {R}^N,\\&[u]_{s,p}=\left( \int _{\mathbb {R}^{N}}\int _{\mathbb {R}^N}\frac{|u(x)- u(y)|^p}{|x-y|^{N+sp}}dxdy\right) ^{1/p} \end{aligned}$$

where $a\ge 0, b>0$, $0<s<\min \{1,N/2p\}$, $2sp\le \mu <N$, $(-\Delta )_p^s$ is the fractional p-Laplace operator, $\lambda >0$ is a parameter, $p_{\mu ,s}^*=\frac{(N-\frac{\mu }{2})p}{N-sp}$ is the critical exponent in the sense of the Hardy–Littlewood–Sobolev inequality, $1<q<p_s^*=\frac{Np}{N-sp}$ and $h\in L^{\frac{p_s^*}{p_s^*-q}}(\mathbb {R}^N)$. Under some suitable assumptions, we obtain the multiplicity of nontrivial solutions by using variational methods. In particular, we get the existence of infinitely many nontrivial solutions for the degenerate Kirchhoff case by using Krasnoselskii’s genus theory.