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Morrey global bounds for singular quasilinear equation below the natural exponent. (arXiv:1810.09271v1 [math.AP])
来源于:arXiv
The aim of this paper is to present the global bounds for renormalized
solutions to the following quasilinear elliptic problem: \begin{align*}
\begin{cases} -\div(A(x,\nabla u)) &= \mu \quad \text{in} \ \ \Omega, \\ u &=0
\quad \text{on} \ \ \partial \Omega, \end{cases} \end{align*} in Lorentz-Morrey
spaces, where $\Omega \subset \mathbb{R}^n$ ($n \ge 2$), $\mu$ is a finite
Radon measure, $A$ is a monotone Carath\'eodory vector valued function defined
on $W^{1,p}_0(\Omega)$ and the $p$-capacity uniform thickness condition is
imposed on our domain. There have been research activities on the gradient
estimates in Lorentz-Morrey spaces with various hypotheses. For instance, in
\cite{55Ph1} Nguyen Cong Phuc proposed the Morrey global bounds of solution to
this equation, but for the regular case $2-\frac{1}{n}<p\le n$, in
\cite{MP2018}, our first result provides us with the good-$\lambda$ bounds of
solution in Lorentz space for $\frac{3n-2}{2n-1}<p \le 2 - \frac{1}{n}$; and in 查看全文>>