Research article

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

  • 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 $0 \lt s \lt 1 \lt p \lt \infty$, $sp \lt N$, $\lambda \gt 0$ 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.

    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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  • 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 $0 \lt s \lt 1 \lt p \lt \infty$, $sp \lt N$, $\lambda \gt 0$ 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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    © 2021 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
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