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

摘要

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.