Double obstacle problems of nonlinear elliptic equations with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>log</mml:mi> </mml:mrow> </mml:math> -BMO matrix weights^*

Abstract A quasilinear elliptic equation with irregular double obstacles involving almost variable exponent growth is widely extended to a degenerate/singular one by dealing with matrix weights connected to the gradient component in the nonlinearity. An optimal Calderón–Zygmund type estimate for the gradient of a weak solution is obtained in the setting of a weighted Musielak–Orlicz space by finding a minimal regularity assumption on the matrix weight which turns out a smallness in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>log</mml:mi> </mml:mrow> </mml:math> -BMO. To this end, we compare a solution to the underlying solution with constant-matrix equations. In the process, we make a series of comparison estimates starting from a degenerate/singular double obstacle problem to the associated obstacle problem and then to the limiting problem having a desired Lipschitz regularity. Our regularity result provides a newly achieved theory in the realm of highly nonlinear elliptic equations with kinds of nonstandard growth conditions as we deal with such degenerate/singular nonlinearities in terms of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>log</mml:mi> </mml:mrow> </mml:math> -BMO instead of BMO.