Infinitely many solutions for Kirchhoff-type variable-order fractional Laplacian problems involving variable exponents

ABSTRACT

In this paper, we show the existence of infinitely many solutions for Kirchhoff-type variable-order fractional Laplacian problems involving variable exponents. More precisely, we consider $\left\{\begin{array}{ll}M\left(\right[u{]}_{s(\cdot )}^2\left)\right(-\mathrm{\Delta }{)}^{s(\cdot )}u+V\left(x\right)u=\lambda |u{|}^{p\left(x\right)-2}u+\mu |u{|}^{q\left(x\right)-2}u& \mathrm{i}\mathrm{n}\mathrm{\Omega },\\ u=0& \mathrm{i}\mathrm{n}{\mathbb{R}}^N\setminus \mathrm{\Omega },\end{array}\right.$ where $[u{]}_{s(\cdot )}:={\left({\iint }_{{\mathbb{R}}^{2N}}\frac{|u\left(x\right)-u\left(y\right){|}^2}{|x-y{|}^{N+2s(x,y)}}\mathrm{d}x\mathrm{d}y\right)}^{1/2},$$N\ge 1,s(\cdot ):{\mathbb{R}}^N\times {\mathbb{R}}^N\to (0,1)$ is a continuous function, Ω is a bounded domain in ${\mathbb{R}}^N$ with $N>2s(x,y)$ for all $(x,y)\in \mathrm{\Omega }\times \mathrm{\Omega },(-\mathrm{\Delta }{)}^{s(\cdot )}$ is the variable-order fractional Laplace operator, $M:{\mathbb{R}}_0^{+}\to {\mathbb{R}}_0^{+}$ and $V:\mathrm{\Omega }\to [0,\mathrm{\infty })$ are two continuous functions, $\alpha ,\text{β}>0$ are two parameters and $p,q\in C\left(\mathrm{\Omega }\right).$ In addition to using the new version of Clark's theorem due to Liu and Wang to prove the existence of infinitely many solutions for the above problem, we also apply the symmetric mountain pass theorem, fountain theorem and dual fountain theorem to obtain the same conclusion. The main feature, as well as the main difficulty, of our problem is the fact that the Kirchhoff term M could be zero at zero.

Keywords:

• Kirchhoff-type problem
• variable-order fractional Laplacian

2010 Mathematics Subject Classifications:

• 35R11
• 35A15
• 35J60
• 47G20

Acknowledgments

The authors would like to thank Professor Giovanni Molica Bisci for valuable comments and suggestions on improving this paper.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

Li Wang was supported by National Natural Science Foundation of China (Nos. 11701178, 11561024). Binlin Zhang was supported by National Natural Science Foundation of China (No. 11871199).