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In this paper, we study a class of critical elliptic problems of Kirchhoff type: [a+b(∫R3|∇u|2−μu2|x|2dx)2−α2](−Δu−μu|x|2)=|u|2∗(α)−2u|x|α+λf(x)|u|q−2u|x|β, where a,b>0, μ∈[0,1/4), α,β∈[0,2), and q∈(1,2) are constants and 2∗(α)=6−2α is the Hardy-Sobolev exponent in R3. For a suitable function f(x), we establish the existence of multiple solutions by using the Nehari manifold and fibering maps. Moreover, we regard b>0 as a parameter to obtain the convergence property of solutions for the given problem as b↘0+ by the mountain pass theorem and Ekeland’s variational principle. [ABSTRACT FROM AUTHOR]

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 and ps. 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. [ABSTRACT FROM AUTHOR]