A Nonhomogeneous Fractional p-Kirchhoff Type Problem Involving Critical Exponent in ℝ N

Abstract

Abstract This paper concerns itself with the nonexistence and multiplicity of solutions for the following fractional Kirchhoff-type problem involving the critical Sobolev exponent: [ a + b ⁢ ( ∬ ℝ 2 ⁢ N | u ⁢ ( x ) - u ⁢ ( y ) | p | x - y | N + p ⁢ s ⁢ 𝑑 x ⁢ 𝑑 y ) θ - 1 ] ⁢ ( - Δ ) p s ⁢ u = | u | p s * - 2 ⁢ u + λ ⁢ f ⁢ ( x )   in  ⁢ ℝ N , \Biggl{[}a+b\biggl{(}\iint_{\mathbb{R}^{2N}}\frac{\lvert u(x)-u(y)\rvert^{p}}{% \lvert x-y\rvert^{N+ps}}\,dx\,dy\biggr{)}^{\theta-1}\Biggr{]}(-\Delta)_{p}^{s}% u=\lvert u\rvert^{p_{s}^{*}-2}u+\lambda f(x)\quad\text{in }\mathbb{R}^{N}, where a ≥ 0 {a\kern-1.0pt\geq\kern-1.0pt0} , b > 0 , θ > 1 {b\kern-1.0pt>\kern-1.0pt0,\theta\kern-1.0pt>\kern-1.0pt1} , ( - Δ ) p s {(-\Delta)_{p}^{s}} is the fractional p-Laplacian with 0 < s < 1 {0\kern-1.0pt<\kern-1.0pts\kern-1.0pt<\kern-1.0pt1} and 1 < p < N / s {1\kern-1.0pt<\kern-1.0ptp\kern-1.0pt<\kern-1.0ptN/s} , p s * = N ⁢ p / ( N - p ⁢ s ) {p_{s}^{*}\kern-1.0pt=\kern-1.0ptNp/(N-ps)} is the critical Sobolev exponent, λ ≥ 0 {\lambda\geq 0} is a parameter, and f ∈ L p s * / ( p s * - 1 ) ⁢ ( ℝ N ) ∖ { 0 } {f\in L^{p_{s}^{*}/(p_{s}^{*}-1)}(\mathbb{R}^{N})\setminus\{0\}} is a nonnegative function. When λ = 0 {\lambda=0} , we show that the multiplicity and nonexistence of solutions for the above problem are related with N, θ, s, p, a, and b. When λ > 0 {\lambda>0} , by using Ekeland’s variational principle and the mountain pass theorem, we show that there exists λ * * > 0 {\lambda^{**}>0} such that the above problem admits at least two nonnegative solutions for all λ ∈ ( 0 , λ * * ) {\lambda\in(0,\lambda^{**})} . In the latter case, in order to overcome the loss of compactness, we derive a fractional version of the principle of concentration compactness in the setting of the fractional p-Laplacian.