Multiple solutions for a coupled Kirchhoff system with fractional p-Laplacian and sign-changing weight functions

Abstract

In this paper, we investigate the multiplicity of solutions for Kirchhoff fractional p-Laplacian system in bounded domains: $\left\{\begin{array}{l}{\left(\sum _{i=1}^k[u_i{]}_{s,p}^p\right)}^{\theta -1}(-\mathrm{\Delta }{)}_p^s{u}_j(x)\\ ={\lambda }_j{f}_j\left(x\right)|u_j{|}^{q-2}{u}_j+\sum _{i\ne j}{\beta }_{i,j}h\left(x\right)|u_i{|}^m|u_j{|}^{m-2}{u}_j& \text{in}\mathrm{\Omega },\\ u_j=0& \mathrm{i}\mathrm{n}{\mathbb{R}}^n\setminus \mathrm{\Omega }.\end{array}\right.$ By using the Nehari manifold method, together with Ekeland's variational principle, we show that there exist two distinct solutions under suitable conditions on weight functions $f_j$ and h. Our results extend and generalize the main results in Xiang et al. [Multiplicity of solutions for a class of quasilinear Kirchhoff system involving the fractional p-Laplacian. Nonlinearity. 2016;29:3186–3205] in Nonlinearity 2016, in which $f_j,h$ are constants.

Keywords:

• Fractional p-Kirchhoff system
• Nehari manifold
• sign-changing weight functions

AMS Subject Classifications:

• 35A15
• 47G20
• 35R11

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

M. Yang was supported by the National Natural Science Foundation of China [grant number 11971184].