## Abstract

Consider the following nonlinear elliptic equation of p(x)-Laplacian type with nonstandard growth {diva(Du,x)=μ in Ω, u=0 on ∂Ω, where Ω is a Reifenberg domain in R^{n}, μ is a Radon measure defined on Ω with finite total mass and the nonlinearity a: R^{n}× R^{n}→ R^{n} is modeled upon the p(·) -Laplacian. We prove the estimates on weighted variable exponent Lebesgue spaces for gradients of solutions to this equation in terms of Muckenhoupt–Wheeden type estimates. As a consequence, we obtain some new results such as the weighted L^{q}- L^{r} regularity (with constants q < r) and estimates on Morrey spaces for gradients of the solutions to this non-linear equation.

Original language | English |
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Article number | 28 |

Pages (from-to) | 1-37 |

Number of pages | 37 |

Journal | Nonlinear Differential Equations and Applications |

Volume | 25 |

Issue number | 4 |

DOIs | |

Publication status | Published - 1 Aug 2018 |

## Keywords

- Measure data
- Nonlinear p(x)-Laplacian type equation
- Reifenberg domain
- Weighted generalized Lebesgue spaces