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Liouville type theorems, a priori estimates and existence of solutions for sub-critical order Lane—Emden—Hardy equations

Abstract

We study the sub-critical order Lane—Emden—Hardy equations

$${\left( { - {\rm{\Delta }}} \right)^m}u\left( x \right) = {{{u^p}\left( x \right)} \over {{{\left| x \right|}^a}}}\;\;\;\;{\rm{in}}\;{\mathbb{R}^n}$$
((0.1))

with n ≥ 3, \(1 \le m < {n \over 2}\), 0 ≤ a < 2m and p > 1. We establish Liouville theorems in the ranges \(1 < p < {{n + 2m - 2a} \over {n - 2m}}\) if 0 ≤ a < 2 and 1 < p < +∞ if 2 ≤ a < 2m for nonnegative classical solutions of equations (0.1), that is, the unique nonnegative solution is u ≡ 0. As an application, we derive a priori estimates and the existence of positive solutions to sub-critical order Lane—Emden equations in bounded domains.

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Acknowledgements

The authors are grateful to the referees for their careful reading and valuable comments and suggestions that improved the presentation of the paper.

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Correspondence to Wei Dai.

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Wei Dai is supported by the NNSF of China (No. 11971049 and 11501021), the Fundamental Research Funds for the Central Universities and the State Scholarship Fund of China (No. 201806025011).

Shaolong Peng is supported by the NNSF of China (No. 11971049).

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Dai, W., Peng, S. & Qin, G. Liouville type theorems, a priori estimates and existence of solutions for sub-critical order Lane—Emden—Hardy equations. JAMA 146, 673–718 (2022). https://doi.org/10.1007/s11854-022-0207-6

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  • DOI: https://doi.org/10.1007/s11854-022-0207-6