Positive and negative solutions for the nonlinear fractional Kirchhoff equation in $${\mathbb {R}}^{N}$$ R N

Yang Wang () and Yansheng Liu ()
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Yang Wang: Shandong Management University
Yansheng Liu: Shandong Normal University

Partial Differential Equations and Applications, 2020, vol. 1, issue 5, 1-19

Abstract: Abstract This paper is concerned with the following nonlinear fractional Kirchhoff equation $$\begin{aligned} (a+\lambda \int _{{\mathbb {R}}^{N}}|(-\varDelta )^{\frac{s}{2}}u|^{2}dx)(-\varDelta )^{s}u+V(x)u=f(x,u)+ w(x)|u|^{q-2}u,\ \ \ x\in {\mathbb {R}}^{N}, \end{aligned}$$ ( a + λ ∫ R N | ( - Δ ) s 2 u | 2 d x ) ( - Δ ) s u + V ( x ) u = f ( x , u ) + w ( x ) | u | q - 2 u , x ∈ R N , where $$N>2s,\ a>0, \lambda \ge 0$$ N > 2 s , a > 0 , λ ≥ 0 is a parameter, $$(-\varDelta )^{s}$$ ( - Δ ) s denotes the fractional Laplacian operator of order $$s\in (0, 1),\ 2_{s}^{\star }=\frac{2N}{N-2s},\ V$$ s ∈ ( 0 , 1 ) , 2 s ⋆ = 2 N N - 2 s , V and f are continuous, and $$w(x)\in L^{\frac{2_{s}^{\star }}{2_{s}^{\star }-q}}({\mathbb {R}}^{N}, {\mathbb {R}}^{+})$$ w ( x ) ∈ L 2 s ⋆ 2 s ⋆ - q ( R N , R + ) with $$1

Keywords: Fractional Kirchhoff equation; Variational method; Pohozaev identity; Ekeland’s variational principle; 35J20; 35J60 (search for similar items in EconPapers)
Date: 2020
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DOI: 10.1007/s42985-020-00030-2

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