Existence of nontrivial solutions for Schrödinger-Kirchhoff type equations involving the fractional <i>p</i>-Laplacian and local nonlinearity

Liu Gao; Chunfang Chen; Jianhua Chen; Chuanxi Zhu; Liu Gao; Chunfang Chen; Jianhua Chen; Chuanxi Zhu

doi:10.3934/math.2021083

AIMS Mathematics

2021, Volume 6, Issue 2: 1332-1347. doi: 10.3934/math.2021083

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Research article

Existence of nontrivial solutions for Schrödinger-Kirchhoff type equations involving the fractional p-Laplacian and local nonlinearity

Department of Mathematics, Nanchang University, Nanchang, Jiangxi, 330031, P. R. China

Received: 10 September 2020 Accepted: 09 November 2020 Published: 17 November 2020
MSC : 35R11, 35J60, 35J20

In this paper, we deal with the existence of nontrivial solutions for the following Kirchhoff-type equation $ M\left(\,\,\displaystyle{\iint_{\mathbb{R}^{2N}}\frac{|u(x)-u(y)|^{p}}{|x-y|^{N+sp}}\text{d}x\text{d}y}\right)(-\Delta)_{p}^{s} u+V(x)|u|^{p-2}u = \lambda f(x,u),\,\, \text{in}\,\,\mathbb{R}^N, $ where $0 < s < 1 < p < \infty$, $sp < N$, $\lambda > 0$ is a real parameter, $(-\Delta)_{p}^{s}$ is the fractional $p$-Laplacian operator, $V:\mathbb{R}^N\rightarrow\mathbb{R}^N$ is a potential function, $M$ is a Kirchhoff function, the nonlinearity $f:\mathbb{R}^N\times\mathbb{R}\rightarrow\mathbb{R}$ is a continuous function and just super-linear in a neighborhood of $u = 0$. By using an appropriate truncation argument and the mountain pass theorem, we prove the existence of nontrivial solutions for the above equation, provided that $\lambda$ is sufficiently large. Our results extend and improve the previous ones in the literature.
- fractional p-Laplacian,
- Kirchhoff equations,
- local nonlinearity,
- Moser iteration method,
- truncation argument
Citation: Liu Gao, Chunfang Chen, Jianhua Chen, Chuanxi Zhu. Existence of nontrivial solutions for Schrödinger-Kirchhoff type equations involving the fractional p-Laplacian and local nonlinearity[J]. AIMS Mathematics, 2021, 6(2): 1332-1347. doi: 10.3934/math.2021083

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Abstract

In this paper, we deal with the existence of nontrivial solutions for the following Kirchhoff-type equation $ M\left(\,\,\displaystyle{\iint_{\mathbb{R}^{2N}}\frac{|u(x)-u(y)|^{p}}{|x-y|^{N+sp}}\text{d}x\text{d}y}\right)(-\Delta)_{p}^{s} u+V(x)|u|^{p-2}u = \lambda f(x,u),\,\, \text{in}\,\,\mathbb{R}^N, $ where $0 < s < 1 < p < \infty$, $sp < N$, $\lambda > 0$ is a real parameter, $(-\Delta)_{p}^{s}$ is the fractional $p$-Laplacian operator, $V:\mathbb{R}^N\rightarrow\mathbb{R}^N$ is a potential function, $M$ is a Kirchhoff function, the nonlinearity $f:\mathbb{R}^N\times\mathbb{R}\rightarrow\mathbb{R}$ is a continuous function and just super-linear in a neighborhood of $u = 0$. By using an appropriate truncation argument and the mountain pass theorem, we prove the existence of nontrivial solutions for the above equation, provided that $\lambda$ is sufficiently large. Our results extend and improve the previous ones in the literature.

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