Weighted Lorentz Gradient Estimates for a Class of Quasilinear Elliptic Equations with Measure Data

Abstract

In this paper, we obtain the following local weighted Lorentz gradient estimates

$$\begin{aligned} g^{-1}\left( {\mathcal {M}}_1(\mu ) \right) \in L_{w,{\text {loc}}}^{q,r}(\Omega ) \Longrightarrow |Du| \in L_{w,{\text {loc}}}^{q,r}(\Omega ) \end{aligned}$$

for the weak solutions of a class of non-homogeneous quasilinear elliptic equations with measure data

$$\begin{aligned} -\text {div} ~\! \left( a\left( \left| \nabla u \right| \right) \nabla u \right) = \mu , \end{aligned}$$

where $g(t)= t a(t)$ for $t\ge 0$ and

$$\begin{aligned} {\mathcal {M}}_1(\mu )(x):=\sup _{r>0}\frac{r|\mu |(B_r(x))}{|B_r(x)|}, \quad x\in {\mathbb {R}}^{n}. \end{aligned}$$

Moreover, we remark that two natural and simple examples of functions g(t) in this work are

$$\begin{aligned} g(t)=t^{p-1} ~~(p\text{-Laplace } \text{ equation) }~~~~~~ \text{ and } ~~~~~~ g(t)=t^{p-1 }\log ^\alpha \big ( 1+t\big ) \quad \text{ for }~\alpha > 0. \end{aligned}$$

Actually, the more general and interesting example is related to (p, q)-growth condition by appropriate gluing of the monomials. We remark that our results improve the known results for such equations.