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