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

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, where a, b> 0 , θ= (N− ps/ 2) / (N− ps) and q∈ (1 , p) are constants, and (−Δ)ps is the fractional p-Laplacian operator with 0 < s< 1 < p< ∞ and ps< N. For suitable f(x) , the above equation possesses at least two nontrivial solutions by variational method for any a, b> 0. Moreover, we regard a> 0 and b> 0 as parameters to obtain convergent properties of solutions for the given problem as a↘ 0 and b↘ 0 , respectively.