The existence of a nontrivial solution to an elliptic equation with critical Sobolev exponent and a general potential well

Ye Xue; Yongzhen Ge; Yunlan Wei; Ye Xue; Yongzhen Ge; Yunlan Wei

doi:10.3934/math.2025336

AIMS Mathematics

2025, Volume 10, Issue 3: 7339-7354. doi: 10.3934/math.2025336

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Research article

The existence of a nontrivial solution to an elliptic equation with critical Sobolev exponent and a general potential well

1.
Department of Mathematics, Changzhou University, Changzhou 213164, China
2.
School of Mathematics and Statistics, Linyi University, Linyi 276000, China

Received: 17 January 2025 Revised: 14 March 2025 Accepted: 21 March 2025 Published: 28 March 2025
MSC : 35A15, 35J60

The purpose of this paper is to examine a class of elliptic problems that involve negative potentials $ a\in L^{\frac{N}{2} }(\Omega) $ and critical nonlinearities. To discuss this, the well-known eigenvalue problem $ -\Delta- a $ is considered. Under some mild assumptions, an existence result is obtained, which extends the existing results to the critical case.
- elliptic equation,
- existence,
- general potential,
- critical Sobolev exponent
Citation: Ye Xue, Yongzhen Ge, Yunlan Wei. The existence of a nontrivial solution to an elliptic equation with critical Sobolev exponent and a general potential well[J]. AIMS Mathematics, 2025, 10(3): 7339-7354. doi: 10.3934/math.2025336

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Abstract

The purpose of this paper is to examine a class of elliptic problems that involve negative potentials $ a\in L^{\frac{N}{2} }(\Omega) $ and critical nonlinearities. To discuss this, the well-known eigenvalue problem $ -\Delta- a $ is considered. Under some mild assumptions, an existence result is obtained, which extends the existing results to the critical case.

References

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