Calculus of Variations and Geometric Measure Theory

Q. H. Nguyen - C. P. Nguyen

Existence and regularity estimates for quasilinear equations with measure data: the case $1<p\leq \frac{3n-2}{2n-1}$

created by nguyen on 20 Apr 2021


Accepted Paper

Inserted: 20 apr 2021
Last Updated: 20 apr 2021

Journal: Analysis and PDEs
Year: 2021

ArXiv: 2003.03725 PDF


We obtain existence and global regularity estimates for gradients of solutions to quasilinear elliptic equations with measure data whose prototypes are of the form $-div (
\nabla u
^{p-2} \nabla u)= \delta\,
\nabla u
^q +\mu$ in a bounded main $\Omega\subset\mathbb{R}^n$ potentially with non-smooth boundary. Here either $\delta=0$ or $\delta=1$, $\mu$ is a finite signed Radon measure in $\Omega$, and $q$ is of linear or super-linear growth, i.e., $q\geq 1$. Our main concern is to extend earlier results to the strongly singular case $1<p\leq \frac{3n-2}{2n-1}$. In particular, in the case $\delta=1$ which corresponds to a Riccati type equation, we settle the question of solvability that has been raised for some time in the literature.