Least energy sign-changing solution for a fractional $ p $-Laplacian problem with exponential critical growth

Kun Cheng; Wentao Huang; Li Wang; Kun Cheng; Wentao Huang; Li Wang

doi:10.3934/math.20221140

AIMS Mathematics

2022, Volume 7, Issue 12: 20797-20822. doi: 10.3934/math.20221140

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Research article

Least energy sign-changing solution for a fractional $ p $-Laplacian problem with exponential critical growth

1.
School of Information Engineering, Jingdezhen Ceramic University, Jingdezhen 333403, China
2.
College of Science, East China Jiaotong University, Nanchang 330013, China

Received: 21 August 2022 Revised: 14 September 2022 Accepted: 21 September 2022 Published: 27 September 2022
MSC : 35J60, 35J92, 35R11

In this paper, we consider the following fractional $ p $-Laplacian equation involving Trudinger-Moser nonlinearity:

$ (-\Delta)_{N/s}^{s} u+V(x)|u|^{\frac{N}{s}-2} u = f(u) \ \ {\rm { in }}\ \mathbb{R}^{N}, $

where $ s \in(0, 1), 2 < \frac{N}{s} = p $. The nonlinear function $ f $ has exponential critical growth, and potential $ V $ is a continuous function. By using the constrained variational methods, quantitative Deformation Lemma and Brouwer degree theory, we prove the existence of least energy sign-changing solutions.
- sign-changing solution,
- fractional $ p $-Laplacian,
- exponential critical growth
Citation: Kun Cheng, Wentao Huang, Li Wang. Least energy sign-changing solution for a fractional $ p $-Laplacian problem with exponential critical growth[J]. AIMS Mathematics, 2022, 7(12): 20797-20822. doi: 10.3934/math.20221140

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Abstract

In this paper, we consider the following fractional $ p $-Laplacian equation involving Trudinger-Moser nonlinearity:

$ (-\Delta)_{N/s}^{s} u+V(x)|u|^{\frac{N}{s}-2} u = f(u) \ \ {\rm { in }}\ \mathbb{R}^{N}, $

where $ s \in(0, 1), 2 < \frac{N}{s} = p $. The nonlinear function $ f $ has exponential critical growth, and potential $ V $ is a continuous function. By using the constrained variational methods, quantitative Deformation Lemma and Brouwer degree theory, we prove the existence of least energy sign-changing solutions.

References

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