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 [ M 1 ( μ ) ] ∈ L w , l o c q , r ( Ω ) ⟹ | D u | ∈ L w , l o c q , r ( Ω ) , where g ( t ) = t a ( t ) for t ≥ 0 and M 1 ( μ ) ( x ) is the first-order fractional maximal function M 1 ( μ ) ( x ) := sup r > 0 r | μ | ( B r ( x ) ) | B r ( x ) | , for a class of non-homogeneous divergence quasilinear elliptic equations with measure data in the subquadratic case − div ⁡ [ a ( ( A D u ⋅ D u ) 1 2 ) A D u ] = μ in Ω , whose model cases are the classical elliptic p -Laplacian equation with measure data − div ⁡ ( | D u | p − 2 D u ) = μ for 1 < p < 2 and the elliptic p -Laplacian equation with the logarithmic term and measure data − div ⁡ ( | D u | p − 2 D u ) = μ for 1 < p < 2 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 | D u | → 0 for 1 < p < 2 .