Lipschitz regularity for fully nonlinear elliptic equations with ( p , q )-growth

We prove the interior and global Lipschitz regularity results for a solution of fully nonlinear equations with <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>p</m:mi> <m:mo>,</m:mo> <m:mi>q</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> {(p,q)} -growth. We prove that for a small gap <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>q</m:mi> <m:mo>-</m:mo> <m:mi>p</m:mi> </m:mrow> </m:math> {q-p} , a solution is locally or globally Lipschitz continuous. We also prove that a given Hölder continuous solution is Lipschitz continuous under improved bounds for the gap. These gap conditions are similar to those required for the regularity of double phase problems in divergence form as in [P. Baroni, M. Colombo and G. Mingione, Regularity for general functionals with double phase, Calc. Var. Partial Differential Equations 57 2018, 2, Paper No. 62] and [M. Colombo and G. Mingione, Bounded minimisers of double phase variational integrals, Arch. Ration. Mech. Anal. 218 2015, 1, 219–273].