论文标题
存在椭圆形问题的积极解决方案，并具有关键生长和对数扰动
The existence of positive solution for an elliptic problem with critical growth and logarithmic perturbation
论文作者
论文摘要
我们认为，对于以下brézis-nirenberg问题，积极解决方案的存在和不存在与对数扰动的问题： \ begin {equation*} \ begin {case} -ΔU= {\ left | u \ right |}^{{2}^{\ ast} -2} u+λu+λu+μu+μu\ log {u}^{2}^{2}＆x \ inω， \ quad \; \：\，u = 0＆x \ in \partialΩ， \ end {cases} \ end {equation*} 其中$ω$ $ \ subset $ $ \ r^n $是一个有界的光滑域，$λ，μ\ in \ r $，$ n \ ge3 $和$ {2}^{\ ast}：= \ frac {2n} {n-2} {n-2} {n-2} {n-2} {n-2} {n-2} {n-2} {n-2} {n-2} { l^{2^\ ast}（ω）$。 $ s \ log s^2 $ in $（0， +\ infty）$的符号的不确定性本身具有一定的兴趣。我们将展示存在的阳性基态解决方案的存在，该解决方案的山通道类型提供了$λ\ in \ r，μ> 0 $和$ n \ geq 4 $。 $μ<0 $的情况很刺。但是，对于$ n = 3,4 $ $ $λ\ in（ - \ infty，λ_1（ω））$，我们还可以在一些进一步合适的假设下建立积极解决方案的存在。并且还获得了$μ<0 $和$ - \ frac {（n-2）μ} {2}+\ frac {（n-2）μ} {2} {2} \ log（ - \ frac {（（N-2）μ} {2} {2} {2} {2} {2} {2} {2} {2} {2} {2} {2}）+λ-λ_1（gee）$ n $ n $ n $ n $ geq 0与Brézis，H。和Nirenberg，L。的结果相比（Comm。PureAppl。Math。1983），当对数扰动上的参数$μ$不为零时，就会发生一些新的有趣现象。
We consider the existence and nonexistence of positive solution for the following Brézis-Nirenberg problem with logarithmic perturbation: \begin{equation*} \begin{cases} -Δu={\left|u\right|}^{{2}^{\ast }-2}u+λu+μu\log {u}^{2} &x\in Ω, \quad \;\:\, u=0& x\in \partial Ω, \end{cases} \end{equation*} where $Ω$ $\subset$ $\R^N$ is a bounded smooth domain, $λ, μ\in \R$, $N\ge3$ and ${2}^{\ast }:=\frac{2N}{N-2}$ is the critical Sobolev exponent for the embedding $H^1_{0}(Ω)\hookrightarrow L^{2^\ast}(Ω)$. The uncertainty of the sign of $s\log s^2$ in $(0, +\infty)$ has some interest in itself. We will show the existence of positive ground state solution which is of mountain pass type provided $λ\in \R, μ>0$ and $N\geq 4$. While the case of $μ<0$ is thornier. However, for $N=3,4$ $λ\in (-\infty, λ_1(Ω))$, we can also establish the existence of positive solution under some further suitable assumptions. And a nonexistence result is also obtained for $μ<0$ and $-\frac{(N-2)μ}{2}+\frac{(N-2)μ}{2}\log(-\frac{(N-2)μ}{2})+λ-λ_1(Ω)\geq 0$ if $N\geq 3$. Comparing with the results in Brézis, H. and Nirenberg, L. (Comm. Pure Appl. Math. 1983), some new interesting phenomenon occurs when the parameter $μ$ on logarithmic perturbation is not zero.
