  We give a simple proof of the strong maximum principle for viscosity
subsolutions of fully nonlinear elliptic PDEs on the form $$ F(x,u,Du,D^2u) = 0
$$ under suitable structure conditions on the equation allowing for
non-Lipschitz growth in the gradient terms. In case of smooth boundaries, we
also prove the Hopf lemma, the boundary Harnack inequality and that positive
viscosity solutions vanishing on a portion of the boundary are comparable with
the distance function near the boundary. Our results apply to weak solutions of
an eigenvalue problem for the variable exponent $p$-Laplacian.
