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The viscosity solutions of a nonlinear equation related to the p-Laplacian

The viscosity solutions of a nonlinear equation related to the p-Laplacian are considered. Besides there is a damping term in the equation, a nonlocal function is added. By considering the regularized problem and using Moser iteration technique, we get the uniformly local bounded properties of the solutions and the L p -norm for the gradients. By the compactness theorem, we prove the existence of the viscosity solution of the equation.