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Volume 35, Issue 4
Positive Ground State Solutions for a Critical Nonlocal Problem in Dimension Three

Xiaotao Qian

DOI: 10.4208/jpde.v35.n4.6

J. Part. Diff. Eq., 35 (2022), pp. 382-394.

Published online: 2022-10

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  • Abstract

In this paper, we are interested in the following nonlocal problem with critical exponent \begin{align*} \begin{cases} -\left(a-b\displaystyle\int_{\Omega}|\nabla u|^2{\rm d}x\right)\Delta u=\lambda |u|^{p-2}u+|u|^{4}u, &\quad x\in\Omega,\\ u=0,  &\quad x\in\partial\Omega, \end{cases} \end{align*} where $a,b$ are positive constants, $2<p<6$, $\Omega$ is a smooth bounded domain in $\mathbb{R}^3$ and $\lambda>0$ is a parameter. By variational methods, we prove that problem has a positive ground state solution $u_b$ for $\lambda>0$ sufficiently large. Moreover, we take $b$ as a parameter and study the asymptotic behavior of $u_b$ when $b\searrow0$.

  • Keywords

Nonlocal problem, critical exponent, positive solutions, variational methods.

  • AMS Subject Headings

35J20, 35J60

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

qianxiaotao1984@163.com (Xiaotao Qian)

  • BibTex
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  • TXT
@Article{JPDE-35-382, author = {Qian , Xiaotao}, title = {Positive Ground State Solutions for a Critical Nonlocal Problem in Dimension Three}, journal = {Journal of Partial Differential Equations}, year = {2022}, volume = {35}, number = {4}, pages = {382--394}, abstract = {

In this paper, we are interested in the following nonlocal problem with critical exponent \begin{align*} \begin{cases} -\left(a-b\displaystyle\int_{\Omega}|\nabla u|^2{\rm d}x\right)\Delta u=\lambda |u|^{p-2}u+|u|^{4}u, &\quad x\in\Omega,\\ u=0,  &\quad x\in\partial\Omega, \end{cases} \end{align*} where $a,b$ are positive constants, $2<p<6$, $\Omega$ is a smooth bounded domain in $\mathbb{R}^3$ and $\lambda>0$ is a parameter. By variational methods, we prove that problem has a positive ground state solution $u_b$ for $\lambda>0$ sufficiently large. Moreover, we take $b$ as a parameter and study the asymptotic behavior of $u_b$ when $b\searrow0$.

}, issn = {2079-732X}, doi = {https://doi.org/10.4208/jpde.v35.n4.6}, url = {http://global-sci.org/intro/article_detail/jpde/21055.html} }
TY - JOUR T1 - Positive Ground State Solutions for a Critical Nonlocal Problem in Dimension Three AU - Qian , Xiaotao JO - Journal of Partial Differential Equations VL - 4 SP - 382 EP - 394 PY - 2022 DA - 2022/10 SN - 35 DO - http://doi.org/10.4208/jpde.v35.n4.6 UR - https://global-sci.org/intro/article_detail/jpde/21055.html KW - Nonlocal problem, critical exponent, positive solutions, variational methods. AB -

In this paper, we are interested in the following nonlocal problem with critical exponent \begin{align*} \begin{cases} -\left(a-b\displaystyle\int_{\Omega}|\nabla u|^2{\rm d}x\right)\Delta u=\lambda |u|^{p-2}u+|u|^{4}u, &\quad x\in\Omega,\\ u=0,  &\quad x\in\partial\Omega, \end{cases} \end{align*} where $a,b$ are positive constants, $2<p<6$, $\Omega$ is a smooth bounded domain in $\mathbb{R}^3$ and $\lambda>0$ is a parameter. By variational methods, we prove that problem has a positive ground state solution $u_b$ for $\lambda>0$ sufficiently large. Moreover, we take $b$ as a parameter and study the asymptotic behavior of $u_b$ when $b\searrow0$.

Xiaotao Qian. (2022). Positive Ground State Solutions for a Critical Nonlocal Problem in Dimension Three. Journal of Partial Differential Equations. 35 (4). 382-394. doi:10.4208/jpde.v35.n4.6
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