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We establish existence results for a class of mixed anisotropic and nonlocal  p -Laplace equations with singular nonlinearities. We consider both constant and variable singular exponents. Our argument is based on an approximation method. To this end, we also discuss the necessary regularity properties of weak solutions of the associated non-singular problems. More precisely, we obtain local boundedness of subsolutions, the Harnack inequality for solutions and the weak Harnack inequality for supersolutions.

On the regularity theory for mixed anisotropic and nonlocal p -Laplace equations and its applications to singular problems