Volume 25, Issue 2
Existence of Positive Solutions for Kirchhoff Type Problems with Critical Exponent

J. Part. Diff. Eq., 25 (2012), pp. 187-198.

Published online: 2012-06

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• Abstract

In this paper,we consider the following Kirchhoff type problemwith critical exponent  $-(a+b∫_Ω|∇u|^2dx)Δu=λu^q+u^5, in\ Ω, u=0, on\ ∂Ω$,  where $Ω⊂R^3$ is a bounded smooth domain, $0< q < 1$ and the parameters $a,b,λ > 0$. We show that there exists a positive constant $T_4(a)$ depending only on a, such that for each $a > 0$ and $0 < λ < T_4(a)$, the above problem has at least one positive solution. The method we used here is based on the Nehari manifold, Ekeland's variational principle and the concentration compactness principle.

• Keywords

Freedricksz transition variational problem liquid crystals Landau-de Gennes functional

35J60, 35B33

yjsun@gucas.ac.cn (Yijing Sun)

liuxingcas@gmail.com (Xing Liu)

• BibTex
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@Article{JPDE-25-187, author = {Sun , Yijing and Liu , Xing}, title = {Existence of Positive Solutions for Kirchhoff Type Problems with Critical Exponent}, journal = {Journal of Partial Differential Equations}, year = {2012}, volume = {25}, number = {2}, pages = {187--198}, abstract = {

In this paper,we consider the following Kirchhoff type problemwith critical exponent  $-(a+b∫_Ω|∇u|^2dx)Δu=λu^q+u^5, in\ Ω, u=0, on\ ∂Ω$,  where $Ω⊂R^3$ is a bounded smooth domain, $0< q < 1$ and the parameters $a,b,λ > 0$. We show that there exists a positive constant $T_4(a)$ depending only on a, such that for each $a > 0$ and $0 < λ < T_4(a)$, the above problem has at least one positive solution. The method we used here is based on the Nehari manifold, Ekeland's variational principle and the concentration compactness principle.

}, issn = {2079-732X}, doi = {https://doi.org/10.4208/jpde.v25.n2.5}, url = {http://global-sci.org/intro/article_detail/jpde/5182.html} }
TY - JOUR T1 - Existence of Positive Solutions for Kirchhoff Type Problems with Critical Exponent AU - Sun , Yijing AU - Liu , Xing JO - Journal of Partial Differential Equations VL - 2 SP - 187 EP - 198 PY - 2012 DA - 2012/06 SN - 25 DO - http://doi.org/10.4208/jpde.v25.n2.5 UR - https://global-sci.org/intro/article_detail/jpde/5182.html KW - Freedricksz transition KW - variational problem KW - liquid crystals KW - Landau-de Gennes functional AB -

In this paper,we consider the following Kirchhoff type problemwith critical exponent  $-(a+b∫_Ω|∇u|^2dx)Δu=λu^q+u^5, in\ Ω, u=0, on\ ∂Ω$,  where $Ω⊂R^3$ is a bounded smooth domain, $0< q < 1$ and the parameters $a,b,λ > 0$. We show that there exists a positive constant $T_4(a)$ depending only on a, such that for each $a > 0$ and $0 < λ < T_4(a)$, the above problem has at least one positive solution. The method we used here is based on the Nehari manifold, Ekeland's variational principle and the concentration compactness principle.

Yijing Sun & Xing Liu. (2019). Existence of Positive Solutions for Kirchhoff Type Problems with Critical Exponent. Journal of Partial Differential Equations. 25 (2). 187-198. doi:10.4208/jpde.v25.n2.5
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