Morrey global bounds for singular quasilinear equation below the natural exponent. (arXiv:1810.09271v1 [math.AP])

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>>