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

Abstract

AbstractWe establish the equivalence between weak and viscosity solutions to the nonhomogeneous double phase equation with lower-order term $$\begin{aligned} -{\text {div}}(|Du|^{p-2}Du+a(x)|Du|^{q-2}Du)=f(x,u,Du),\quad 1<p\le q<\infty ,\ a(x)\ge 0. \end{aligned}$$ - div ( | D u | p - 2 D u + a ( x ) | D u | q - 2 D u ) = f ( x , u , D u ) , 1 < p ≤ q < ∞ , a ( x ) ≥ 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 a priori Lipschitz continuity are weak solutions of such equation by virtue of the $$\inf $$ inf ($$\sup $$ 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.