On the regularity theory for mixed local and nonlocal quasilinear elliptic equations

We consider a combination of local and nonlocal p p -Laplace equations discuss several regularity properties weak solutions. More precisely, we establish boundedness subsolutions, Hölder continuity solutions, Harnack inequality for solutions supersolutions. also lower semicontinuity supersolutions as well upper subsolutions. Our approach is purely analytic it based on the De Giorgi-Nash-Moser theory, expansion positivity estimates involving tail term. The main results apply to sign changing capture both features equation.