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 - <p style="text-align: justify;">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, &amp;\quad x\in\Omega,\\ u=0,&nbsp; &amp;\quad x\in\partial\Omega, \end{cases} \end{align*} where $a,b$ are positive constants, $2&lt;p&lt;6$, $\Omega$ is a smooth bounded domain in $\mathbb{R}^3$ and $\lambda&gt;0$ is a parameter. By variational methods, we prove that problem has a positive ground state solution $u_b$ for $\lambda&gt;0$ sufficiently large. Moreover, we take $b$ as a parameter and study the asymptotic behavior of $u_b$ when $b\searrow0$.</p>