We study the interior weighted Sobolev regularity for weak solutions of the quasilinear equations of
the form
divA(x,u,∇u)=divF. The vector
field
A is allowed to
be discontinuous in
x,
Hölder continuous in
u
and its growth in the gradient variable is like the
p-Laplace operator with
1<p<∞. We establish interior
weighted
W1,q-regularity
estimates for weak solutions to the equations for every
q>p
assuming that the weak solutions are in the local John–Nirenberg
BMO
space. This paper therefore improves available results because it replaces the
boundedness or continuity assumption on weak solutions by the borderline
BMO
one. Our regularity estimates also recover known results in which
A is independent
of the variable
u.
Our regularity theory complements the classical
C1,α-regularity
theory developed by many mathematicians including DiBenedetto and Tolksdorf for
this general class of quasilinear elliptic equations.
