论文标题
关于广义的布雷兹 - 尼伦贝格问题
On the generalised Brezis-Nirenberg problem
论文作者
论文摘要
对于$ p \ in（1，n）$和一个域$ω$ in $ \ mathbb {r}^n $，我们研究以下准线性问题，涉及关键增长：\ begin {eqnarray*} -Δ_Pu-μg| u |^{p-2} u = | U |^{p^{*} -2} u \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ mathcal {d} _p（Ω） \ text {div}（| \ nabla u |^{p-2} \ nabla u），$ $ $ p^{*} = \ frac {np} {np} {n-p} $是关键的sobolev exponent和$ \ nathcal {d} $ \ text {c} _c^{\ infty}（ω）$相对于norm $ $ \ | | U \ \ | _ {\ Mathcal {d} {d} _p}：= \ left [\ left [\ left [\ displaystyle \ int_phissystyle \ int_phissy \ int_phiss | \ nabla | \ nabla U |^p \^p \^p \ mathrm}本文，我们在$ g $和$ω$上提供各种足够的条件，以便上述问题对某些$μ$的积极解决方案提供了积极的解决方案。 As a consequence, for $N \geq p^2$, if $g $ is such that $g^+ \neq 0$ and the map $u \mapsto \displaystyle \int_Ω |g||u|^p \mathrm{d}x$ is compact on $\mathcal{D}_p(Ω)$, we show that the problem under consideration has a positive solution for certain range of $μ$。此外，对于$ω= \ mathbb {r}^n $，我们为存在阳性解决方案提供了必要的条件。
For $ p \in (1,N)$ and a domain $Ω$ in $\mathbb{R}^N$, we study the following quasi-linear problem involving the critical growth: \begin{eqnarray*} -Δ_p u - μg|u|^{p-2}u = |u|^{p^{*}-2}u \ \mbox{ in } \mathcal{D}_p(Ω), \end{eqnarray*} where $Δ_p$ is the $p$-Laplace operator defined as $Δ_p(u) = \text{div}(|\nabla u|^{p-2} \nabla u),$ $p^{*}= \frac{Np}{N-p}$ is the critical Sobolev exponent and $\mathcal{D}_p(Ω)$ is the Beppo-Levi space defined as the completion of $\text{C}_c^{\infty}(Ω)$ with respect to the norm $\|u\|_{\mathcal{D}_p} := \left[ \displaystyle \int_Ω |\nabla u|^p \mathrm{d}x \right]^ \frac{1}{p}.$ In this article, we provide various sufficient conditions on $g$ and $Ω$ so that the above problem admits a positive solution for certain range of $μ$. As a consequence, for $N \geq p^2$, if $g $ is such that $g^+ \neq 0$ and the map $u \mapsto \displaystyle \int_Ω |g||u|^p \mathrm{d}x$ is compact on $\mathcal{D}_p(Ω)$, we show that the problem under consideration has a positive solution for certain range of $μ$. Further, for $Ω=\mathbb{R}^N$, we give a necessary condition for the existence of positive solution.
