Multiplicity and asymptotic behavior of solutions to a class of Kirchhoff-type equations involving the fractional p-Laplacian

摘要

Abstract The present study is concerned with the following fractional p-Laplacian equation involving a critical Sobolev exponent of Kirchhoff type: [a+b(∫R2N|u(x)−u(y)|p|x−y|N+psdxdy)θ−1](−Δ)psu=|u|ps∗−2u+λf(x)|u|q−2uin RN, $$iggl[a+b iggl( int_{mathbb {R}^{2N}}rac{|u(x)-u(y)|^{p}}{|x-y|^{N+ps}},dx,dy iggr)^{heta-1} iggr](-Delta)_{p}^{s}u =|u|^{p_{s}^{*}-2}u+lambda f(x)|u|^{q-2}u quadext{in } mathbb {R}^{N}, $$ where a,b>0 $a,b>0$, θ=(N−ps/2)/(N−ps) $heta=(N-ps/2)/(N-ps)$ and q∈(1,p) $qin(1,p)$ are constants, and (−Δ)ps $(-Delta)_{p}^{s}$ is the fractional p-Laplacian operator with 00$. Moreover, we regard a>0 $a>0$ and b>0 $b>0$ as parameters to obtain convergent properties of solutions for the given problem as a↘0+ $asearrow0^{+}$ and b↘0+ $bsearrow0^{+}$, respectively.