Weighted Lorentz estimates for subquadratic quasilinear elliptic equations with measure data

Abstract

In this work we mainly prove the following interior gradient estimates in weighted Lorentz spaces $$ g^{-1} \left[\mathcal M_1(\mu) \right] \in L^{q, r}_{w, loc} (\Omega) \Longrightarrow |Du| \in L^{q, r}_{w, loc} (\Omega), $$ where $g(t)= t a(t)$ for $t\geq 0$ and $\mathcal{M}_1(\mu)(x)$ is the first-order fractional maximal function $$ \mathcal{M}_1(\mu)(x):=\sup_{r>0}\frac{r|\mu|(B_r(x))}{|B_r(x)|}, $$ for a class of non-homogeneous divergence quasilinear elliptic equations with measure data in the subquadratic case \begin{equation*} -\operatorname{div}\left[a \left( \left( A D u \cdot D u\right)^{\frac{1}{2}} \right)A D u \right] =\mu \quad \mbox{in}~ \Omega, \end{equation*} whose model cases are the classical elliptic $p$-Laplacian equation with measure data \begin{align*} -\operatorname{div} \left( \left| D u \right|^{p-2} D u \right)=\mu \quad \mbox{for } 1<p<2 \end{align*} and the elliptic $p$-Laplacian equation with the logarithmic term and measure data \begin{equation*} -\operatorname{div} \left( \left| D u \right|^{p-2}\log \left(1+ \left| D u \right|\right) D u \right)=\mu \quad \mbox{for } 1<p<2. \end{equation*} It deserves to be specially noted that the subquadratic case is a little different from the superquadratic case since as an example, the modulus of ellipticity in the above-mentioned special cases tends to infinity when $|Du| \rightarrow 0$ for $1<p<2$.

Keywords

• weighted
• lorentz
• gradient
• subquadratic
• quasilinear elliptic
• measure