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We establish the equivalence between weak and viscosity solutions to thenonhomogeneous double phase equation with lower-order term $$ -{\rmdiv}(|Du|^{p-2}Du+a(x)|Du|^{q-2}Du)=f(x,u,Du),\quad 1<p\le q<\infty, a(x)\ge0.$$ 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{\em a priori} Lipschitz continuity are weak solutions of such equation byvirtue of the $\inf$($\sup$)-convolution techniques. The reverse implicationcan be concluded through comparison principles. Moreover, we verify that thebounded viscosity solutions are exactly Lipschitz continuous, which is also ofindependent interest.

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