English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

We consider the existence and nonexistence of positive solution for thefollowing Br\'ezis-Nirenberg problem with logarithmic perturbation:  \begin{equation*}  \begin{cases}  -\Delta u={\left|u\right|}^{{2}^{\ast }-2}u+\lambda u+\mu u\log {u}^{2} &x\in\Omega,  \quad \;\:\, u=0& x\in \partial \Omega,  \end{cases}  \end{equation*}  where $\Omega$ $\subset$ $\R^N$ is a bounded smooth domain, $\lambda, \mu \in\R$, $N\ge3$ and ${2}^{\ast }:=\frac{2N}{N-2}$ is the critical Sobolev exponentfor the embedding $H^1_{0}(\Omega)\hookrightarrow L^{2^\ast}(\Omega)$. Theuncertainty of the sign of $s\log s^2$ in $(0, +\infty)$ has some interest initself. We will show the existence of positive ground state solution which isof mountain pass type provided $\lambda\in \R, \mu>0$ and $N\geq 4$. While thecase of $\mu<0$ is thornier. However, for $N=3,4$ $\lambda\in (-\infty,\lambda_1(\Omega))$, we can also establish the existence of positive solutionunder some further suitable assumptions. And a nonexistence result is alsoobtained for $\mu<0$ and$-\frac{(N-2)\mu}{2}+\frac{(N-2)\mu}{2}\log(-\frac{(N-2)\mu}{2})+\lambda-\lambda_1(\Omega)\geq0$ if $N\geq 3$. Comparing with the results in Br\'ezis, H. and Nirenberg, L.(Comm. Pure Appl. Math. 1983), some new interesting phenomenon occurs when theparameter $\mu$ on logarithmic perturbation is not zero.

The existence of positive solution for an elliptic problem with critical growth and logarithmic perturbation