Schrödinger–Kirchhof-type problems involving the fractional p-Laplacian with exponential growth

Abstract

In this paper, we study the existence of a solution to Schrödinger–Kirchhoff-type problems involving a nonlocal integro-differential operator with the Trudinger–Moser nonlinearity. As a particular case, we consider the following fractional problem: $\begin{array}{rl}& M({\iint }_{{\mathbb{R}}^{2N}}\frac{|u(x)-u(y){|}^p}{|x-y{|}^{N+sp}}\mathrm{d}x\mathrm{d}y+{\int }_{{\mathbb{R}}^N}V(x)|u(x){|}^p\mathrm{d}x)((-\mathrm{\Delta }{)}_p^{s}u(x)\\ & +V(x)|u{|}^{p-2}u)=K(x)f(x,u),\end{array}$ where $M:{\mathbb{R}}_0^{+}\to {\mathbb{R}}_0^{+}$ is a continuous function with some appropriate assumptions, $(-\mathrm{\Delta }{)}_p^s$ is the fractional p-Laplacian, $0<s<1<p<\mathrm{\infty }$ with sp = N, K, V are positive continuous functions satisfying some additional conditions, f is a continuous function on ${\mathbb{R}}^N\times \mathbb{R}$ with exponential growth. By using the mountain pass theorem, we obtain the existence of solutions to the above problem in suitable Sobolev space. A novel feature of our paper is that the above problem may be degenerate, that is, the Kirchhoff function $M(0)=0$.

Keywords:

• Fractional p-Laplacian
• Schrödinger–Kirchhoff-type problem
• Mountain pass theorem
• Trudinger–Moser nonlinearity

2010 Mathematics Subject Classifications:

• 35A15
• 35J60
• 35R11

Disclosure statement

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

Data sharing statements

Data sharing is not applicable to this article as no datasets were generated or analysed during the current study.

Additional information

Funding

B. Zhang was supported by the National Natural Science Foundation of China (Grant No. 11871199 and Grant No. 12171152), the Shandong Provincial Natural Science Foundation, PR China (Grant No. ZR2020MA006), and the Cultivation Project of Young and Innovative Talents in Universities of Shandong Province.