论文标题
Sobolev和Hölder的规律性结果的某些奇异非均匀的准线性问题
Sobolev and Hölder regularity results for some singular nonhomogeneous quasilinear problems
论文作者
论文摘要
本文介绍了以下单数准线性方程的研究： \ begin {equation*} （p）\ left \ {\-Δ_{p} u-Δ__{q} u = f（x）u^{ - δ}，\; u> 0 \ text {in} \; \ om; \; u = 0 \ text {on} \ pa \ om， \正确的。 \ end {equation*} 其中$ \ om $是$ \ mathbb {r}^n $中的一个有界域，带有$ c^2 $ boundard $ \ pa \ om $，$ 1 <q <q <q <q <q <p <p <\ infty $，$ \ de> 0 $ and $ f \ in l^\ infty_ {loc} $ \ textnormal {dist}（x，x，\ pa \ om）^{ - \ ba}，$ $ \ ba \ ge 0 $附近$ \ om $的边界附近。我们证明了$ w^{1，p} _ {loc}（\ om）$及其在$ \ ba <p $的边界附近的行为中存在弱解决方案。因此，我们获得了弱解决方案的最佳Sobolev规律性。通过建立比较原则，我们证明了$ \ ba <2- \ frac {1} {p} $的弱解决方案的独特性。随后，对于$ \ ba \ ge P $，我们证明了不存在的结果。此外，我们证明了弱解决方案梯度的hölder规律性，涉及涉及单数非线性和较低阶段的更通用的准线性方程（请参阅\ eqref {prb}）。这个结果是全新的，具有独立的关注。 除此之外，我们还证明了$β+δ\ geq 1 $的最小弱解决方案的规律性，即使以$ p $ - laplace运营商的价格也没有得到充分回答。
This article deals with the study of the following singular quasilinear equation: \begin{equation*} (P) \left\{ \ -Δ_{p}u -Δ_{q}u = f(x) u^{-δ},\; u>0 \text{ in }\; \Om; \; u=0 \text{ on } \pa\Om, \right. \end{equation*} where $\Om$ is a bounded domain in $\mathbb{R}^N$ with $C^2$ boundary $\pa\Om$, $1< q< p<\infty$, $\de>0$ and $f\in L^\infty_{loc}(\Om)$ is a non-negative function which behaves like $\textnormal{dist}(x,\pa\Om)^{-\ba},$ $\ba\ge 0$ near the boundary of $\Om$. We prove the existence of a weak solution in $W^{1,p}_{loc}(\Om)$ and its behaviour near the boundary for $\ba<p$. Consequently, we obtain optimal Sobolev regularity of weak solutions. By establishing the comparison principle, we prove the uniqueness of weak solution for the case $\ba<2-\frac{1}{p}$. Subsequently, for the case $\ba\ge p$, we prove the non-existence result. Moreover, we prove Hölder regularity of the gradient of weak solution to a more general class of quasilinear equations involving singular nonlinearity as well as lower order terms (see \eqref{Prb}). This result is completely new and of independent interest. In addition to this, we prove Hölder regularity of minimal weak solutions of $(P)$ for the case $β+δ\geq 1$ that has not been fully answered in former contributions even for $p$-Laplace operators.
