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Volume 41, Issue 1
Blowing Up Solutions to Slightly Sub- or Super-Critical Lane-Emden Systems with Neumann Boundary Conditions

Qing Guo & Junyuan Liu

Anal. Theory Appl., 41 (2025), pp. 1-34.

Published online: 2025-04

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  • Abstract

We prove that, for some suitable smooth bounded domain, there exists a solution to the following Neumann problem for the Lane-Emden system:

image.png

where $Ω$ is some smooth bounded domain in $\mathbb{R}^N,$ $N ≥ 4,$ $\mu > 0,$ $α > 0,$ $β > 0$ are constants and $ε\ne 0$ is a small number. We show that there exists a solution to the slightly supercritical problem for $ε > 0,$ and for $ε < 0,$ there also exists a solution to the slightly subcritical problem if the domain is not convex.
Comparing with the single elliptic equations, the challenges and novelty are manifested in the construction of good approximate solutions characterizing the boundary behavior under Neumann boundary conditions, in which process, the selection of the range of nonlinear coupling exponents and the weighted Sobolev spaces requires elaborate discussion.

  • AMS Subject Headings

35J57, 35B25, 35B38

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{ATA-41-1, author = {Guo , Qing and Liu , Junyuan}, title = {Blowing Up Solutions to Slightly Sub- or Super-Critical Lane-Emden Systems with Neumann Boundary Conditions}, journal = {Analysis in Theory and Applications}, year = {2025}, volume = {41}, number = {1}, pages = {1--34}, abstract = {

We prove that, for some suitable smooth bounded domain, there exists a solution to the following Neumann problem for the Lane-Emden system:

image.png

where $Ω$ is some smooth bounded domain in $\mathbb{R}^N,$ $N ≥ 4,$ $\mu > 0,$ $α > 0,$ $β > 0$ are constants and $ε\ne 0$ is a small number. We show that there exists a solution to the slightly supercritical problem for $ε > 0,$ and for $ε < 0,$ there also exists a solution to the slightly subcritical problem if the domain is not convex.
Comparing with the single elliptic equations, the challenges and novelty are manifested in the construction of good approximate solutions characterizing the boundary behavior under Neumann boundary conditions, in which process, the selection of the range of nonlinear coupling exponents and the weighted Sobolev spaces requires elaborate discussion.

}, issn = {1573-8175}, doi = {https://doi.org/10.4208/ata.OA-0026}, url = {http://global-sci.org/intro/article_detail/ata/23956.html} }
TY - JOUR T1 - Blowing Up Solutions to Slightly Sub- or Super-Critical Lane-Emden Systems with Neumann Boundary Conditions AU - Guo , Qing AU - Liu , Junyuan JO - Analysis in Theory and Applications VL - 1 SP - 1 EP - 34 PY - 2025 DA - 2025/04 SN - 41 DO - http://doi.org/10.4208/ata.OA-0026 UR - https://global-sci.org/intro/article_detail/ata/23956.html KW - Lane-Emden system, Neumann problem, blow up solutions, reduction method. AB -

We prove that, for some suitable smooth bounded domain, there exists a solution to the following Neumann problem for the Lane-Emden system:

image.png

where $Ω$ is some smooth bounded domain in $\mathbb{R}^N,$ $N ≥ 4,$ $\mu > 0,$ $α > 0,$ $β > 0$ are constants and $ε\ne 0$ is a small number. We show that there exists a solution to the slightly supercritical problem for $ε > 0,$ and for $ε < 0,$ there also exists a solution to the slightly subcritical problem if the domain is not convex.
Comparing with the single elliptic equations, the challenges and novelty are manifested in the construction of good approximate solutions characterizing the boundary behavior under Neumann boundary conditions, in which process, the selection of the range of nonlinear coupling exponents and the weighted Sobolev spaces requires elaborate discussion.

Guo , Qing and Liu , Junyuan. (2025). Blowing Up Solutions to Slightly Sub- or Super-Critical Lane-Emden Systems with Neumann Boundary Conditions. Analysis in Theory and Applications. 41 (1). 1-34. doi:10.4208/ata.OA-0026
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