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

Abstract

We establish the equivalence between weak and viscosity solutions to the nonhomogeneous double phase equation with lower-order term $$-\mathrm{div}(|Du{|}^{p-2}Du+a(x)|Du{|}^{q-2}Du)=f(x,u,Du),1<p\le q<\infty ,a(x)\ge 0.$$ 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 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.