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

Strong maximum principle and boundary estimates for nonhomogeneous elliptic equations