Year: 2022
Author: Xiaotao Qian
Journal of Partial Differential Equations, Vol. 35 (2022), Iss. 4 : pp. 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 $b\searrow0$.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/jpde.v35.n4.6
Journal of Partial Differential Equations, Vol. 35 (2022), Iss. 4 : pp. 382–394
Published online: 2022-01
AMS Subject Headings:
Copyright: COPYRIGHT: © Global Science Press
Pages: 13
Keywords: Nonlocal problem critical exponent positive solutions variational methods.