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

By Mingqi Xiang, Binlin Zhang, Xia Zhang. 2016

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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)ps{(-\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}, ps*=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∈Lps*/(ps*-1)⁢(ℝN)∖{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.

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/b53fc6cdd4886385d5170f50839c462deb15842c

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Citations in the corpus

Superlinear Schrödinger–Kirchhoff type problems involving the fractional p–Laplacian and critical exponent (2019, background): “We would like to point out that the authors in [34] just obtained the existence of two weak solutions for a variant of problem (1.”