Multiplicity and asymptotic behavior of solutions to fractional (p,q)-Kirchhoff type problems with critical Sobolev-Hardy exponent

Abstract

Let Ω ⊂ ℝN be a bounded domain with smooth boundary and 0 ∈ Ω. For 0 < s < 1, 1 ≤ r < q < p, 0 ≤ α < ps < N and a positive parameter λ, we consider the fractional (p, q)-Laplacian problems involving a critical Sobolev-Hardy exponent. This model comes from a nonlocal problem of Kirchhoff type (α + b[u](θ-1)ps,p) (-Δ)spu + (-Δ)squ = |u|p*s(α)-2u/|x|α + λƒ(x) |u|r-2u/|x|c in Ω, u = 0 in ℝN \ Ω, where α, b > 0, c < sr + N(1 - r/p), θ ∈ (1, p*s(α)/p) and p*s(α) is critical Sobolev-Hardy exponent. For a given suitable ƒ(x), we prove that there are least two nontrivial solutions for small λ, by way of the mountain pass theorem and Ekeland's variational principle. Furthermore, we prove that these two solutions converge to two solutions of the limiting problem as α → 0⁺. For the limiting problem, we show the existence of infinitely many solutions, and the sequence tends to zero when λ belongs to a suitable range.