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Volume 23, Issue 4
Uniqueness of the Weak Extremal Solution to Biharmonic Equation with Logarithmically Convex Nonlinearities

Xue Luo

DOI: 10.4208/jpde.v23.n4.2

J. Part. Diff. Eq., 23 (2010), pp. 315-329.

Published online: 2010-11

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  • Abstract

In this note, we investigate the existence of the minimal solution and the uniqueness of the weak extremal (probably singular) solution to the biharmonic equation Δ^2ω=λg(ω) with Dirichlet boundary condition in the unit ball in R^n, where the source term is logarithmically convex. An example is also given to illustrate that the logarithmical convexity is not a necessary condition to ensure the uniqueness of the extremal solution.

  • Keywords

Biharmonic equation logarithmically convex nonlinearities extremal solution uniqueness

  • AMS Subject Headings

35G30 35J40

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{JPDE-23-315, author = {Xue Luo }, title = {Uniqueness of the Weak Extremal Solution to Biharmonic Equation with Logarithmically Convex Nonlinearities}, journal = {Journal of Partial Differential Equations}, year = {2010}, volume = {23}, number = {4}, pages = {315--329}, abstract = {

In this note, we investigate the existence of the minimal solution and the uniqueness of the weak extremal (probably singular) solution to the biharmonic equation Δ^2ω=λg(ω) with Dirichlet boundary condition in the unit ball in R^n, where the source term is logarithmically convex. An example is also given to illustrate that the logarithmical convexity is not a necessary condition to ensure the uniqueness of the extremal solution.

}, issn = {2079-732X}, doi = {https://doi.org/10.4208/jpde.v23.n4.2}, url = {http://global-sci.org/intro/article_detail/jpde/5237.html} }
TY - JOUR T1 - Uniqueness of the Weak Extremal Solution to Biharmonic Equation with Logarithmically Convex Nonlinearities AU - Xue Luo JO - Journal of Partial Differential Equations VL - 4 SP - 315 EP - 329 PY - 2010 DA - 2010/11 SN - 23 DO - http://doi.org/10.4208/jpde.v23.n4.2 UR - https://global-sci.org/intro/article_detail/jpde/5237.html KW - Biharmonic equation KW - logarithmically convex nonlinearities KW - extremal solution KW - uniqueness AB -

In this note, we investigate the existence of the minimal solution and the uniqueness of the weak extremal (probably singular) solution to the biharmonic equation Δ^2ω=λg(ω) with Dirichlet boundary condition in the unit ball in R^n, where the source term is logarithmically convex. An example is also given to illustrate that the logarithmical convexity is not a necessary condition to ensure the uniqueness of the extremal solution.

Xue Luo . (2010). Uniqueness of the Weak Extremal Solution to Biharmonic Equation with Logarithmically Convex Nonlinearities. Journal of Partial Differential Equations. 23 (4). 315-329. doi:10.4208/jpde.v23.n4.2
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