Equivalence of weak and viscosity solutions for the nonhomogeneous double phase equation

We establish the equivalence between weak and viscosity solutions to the nonhomogeneous double phase equation with lower-order term (Formula presented.) We find some appropriate hypotheses on the coefficient a(x), the exponents p, q and the nonlinear term f to show that the viscosity solutions with a priori Lipschitz continuity are weak solutions of such equation by virtue of the inf(sup)-convolution techniques. The reverse implication can be concluded through comparison principles. Moreover, we verify that the bounded viscosity solutions are exactly Lipschitz continuous, which is also of independent interest.