On existence solution for Schrödinger–Kirchhoff-type equations involving the fractional p-Laplacian in ℝ^N

ABSTRACT

The aim of this paper is to study the existence solution for Schrödinger–Kirchhoff-type equations involving nonlocal p-fractional Laplacian$$\begin{array}{cc}& M\left(\int {\int }_{{\mathbb{R}}^{2N}}{|u(x)-u(y)|}^{p}K(x-y)\mathrm{d}x\mathrm{d}y\right){\mathcal{L}}_p^{s}u(x)\hfill \\ \hfill & +{V(x)|u|}^{p-2}u=\lambda f(x,u)+g(x),\hfill \\ \hfill & M(\int {\int }_{{\mathbb{R}}^{2N}}{|u(x)-u(y)|}^{p}K(x-y)\mathrm{d}x\mathrm{d}y+{\int }_{{\mathbb{R}}^N}V(x){|u(x)|}^p\mathrm{d}x)\hfill \\ \hfill & \times ({\mathcal{L}}_p^{s}u(x)+V(x)|u{|}^{p-2}u)=\lambda f(x,u)+g(x),\hfill \end{array}$$

where $\lambda$ is a real positive parameter, $M:[0,\infty )\to (0,\infty )$ is a continuous function, $K:{\mathbb{R}}^N\backslash \{0\}\to {\mathbb{R}}^{+}$ is a singular kernel function, ${\mathcal{L}}_p^s$ is a nonlocal fractional operator, $0<s<1<p<\infty$ with $sp<N,$ $p_s^{\ast }=Np/(N-ps),$ f is a Carathéodory function on ${\mathbb{R}}^N\times \mathbb{R}$ satisfying the Ambrosetti–Rabinowitz-type condition. Using Mountain Pass Theorem, we obtain the existence of above equations. Our result is a extension the problem studied by Pucci–Xiang–Zhang [1].

KEYWORDS:

• Integrodifferential operators
• Schrödinger–Kirchhoff-type equation
• Mountain Pass Theorem

AMS SUBJECT CLASSIFICATIONS:

• Primary: 35A15
• 35J60
• 35R11

Acknowledgements

The authors wish to express thanks to the referee for reading the manuscript very carefully and making a lot of valuable suggestions and comments towards the improvement of the paper and next works.

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

The research results are sponsored by China/Shandong University International Postdoctoral Exchange Program.