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(integral(R2N) vertical bar u(x) - u(y)vertical bar(p)/vertical bar x-y vertical bar(N broken vertical bar) (ps) dx dy)(theta-1)](-Delta)(p)(s)u = vertical bar u vertical bar(p*-2)(s)u + lambda f(x)vertical bar u vertical bar(q-2)u in R-N, where a, b > 0, theta = (N - ps/2)/(N - ps) and q is an element of(1, p) are constants, and (-Delta)(p)(s) is the fractional p-Laplacian operator with 0 < s < 1 < p < infinity 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 SE arrow 0(+) and b SE arrow 0(+), respectively.