Existence of solutions to Kirchhoff type equations involving the nonlocal p₁& … &p_m fractional Laplacian with critical Sobolev-Hardy exponent

Abstract

The aim of this paper is to study the existence of solutions for Kirchhoff type equations involving the nonlocal $p_1\mathrm{\&}\dots \mathrm{\&}{p}_m$ fractional Laplacian with critical Sobolev-Hardy exponent $\left\{\begin{array}{l}{M}_1\left({\int }_{{\mathbb{R}}^{2N}}\frac{|u\left(x\right)-u\left(y\right){|}^{p_1}}{|x-y{|}^{N+p_1{s}_1}}\mathrm{d}x\mathrm{d}y\right)(-\mathrm{\Delta }{)}_{p_1}^{s_1}u+\cdots \\ +M_m\left({\int }_{{\mathbb{R}}^{2N}}\frac{|u\left(x\right)-u\left(y\right){|}^{p_m}}{|x-y{|}^{N+p_m{s}_m}}\mathrm{d}x\mathrm{d}y\right)(-\mathrm{\Delta }{)}_{p_m}^{s_m}u\\ =f(x,u)+\gamma \frac{|u{|}^{p_s^{\ast }\left(\alpha \right)-2}u}{|x{|}^{\alpha }}+\text{β}\frac{|u{|}^{q-2}u}{|x{|}^{\alpha }}\mathrm{i}\mathrm{n}\mathrm{\Omega },\\ u=0\mathrm{i}\mathrm{n}{\mathbb{R}}^N\setminus \mathrm{\Omega },\end{array}\right.$ where $0<s_m<\cdots <s_1=s<1,$ $1<p_m\le \cdots \le p_1=p<\frac{N}{s},m\ge 1,$ $\text{β},\gamma$ are nonnegative constants and $p_s^{\ast }\left(\alpha \right)=\frac{p(N-\alpha )}{N-sp}\le p_s^{\ast }\left(0\right)$ is called the critical Sobolev-Hardy exponent, $1<q<p,$ $0\le \alpha <ps$. Here $(-\mathrm{\Delta }{)}_r^s,$ with $r\in \{p_1,\dots ,p_m\}$ is the fractional r-Laplace operator. Ω is an open bounded subset of ${\mathbb{R}}^N$ with smooth boundary and $0\in \mathrm{\Omega }$. $M_1,\dots ,M_m$ are continuous functions and f is a Carathéodory function which does not satisfy the Ambrosetti-Rabinowitz condition. By using the Mountain Pass Theorem, we obtain the existence of solutions for the above problem. Furthermore, using Fountain Theorem, we get the existence of infinitely many solutions for the above problem when $\gamma =0$. We also study the existence of two nontrivial solutions for the Kirchhoff type equation involving the fractional p-Laplacian via Morse theory. Finally, we consider the case $N=ps,$ and study a degenerate Kirchhoff equation involving Trudinger-Moser nonlinearity. In our best knowledge, it is the first time our problems are studied in this area.

Keywords:

• Integro-differential operators
• fractional p-Laplace
• Fountain Theorem
• Mountain Pass Theorem
• Morse theory
• Trudinger-Moser nonlinearity

AMS Subject Classifications:

• Primary: 35J60
• 35R11
• 35S15

Acknowledgements

The authors wish to thank the editorial board and referees for a very careful reading of the manuscript, and for pointing out misprints that led to the improvement of the original manuscript.

Disclosure statement

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

Additional information

Funding

Nguyen Van Thin is supported by Ministry of Education and Training of Vietnam under project with the name ‘Weak solutions to some class equations, system of partial differential equations containing fractional p-Laplace and Bessel operator’ and [grant number B2020-TNA-06]. Wei Chen is supported by the Science and Technology Research Program of Chongqing Municipal Education Commission [grant number KJQN202000621], the Fundamental Research Funds of Chongqing University of Posts and Telecommunications [grant number CQUPT:A2018-125] and the Basic and Advanced Research Project of CQCSTC [grant number cstc2019jcyj-msxmX].