Equivalence between distributional and viscosity solutions for the double-phase equation

We investigate the different notions of solutions to the double-phase equation $$-\text{dive}(\mathrm{\mid }Du{\mathrm{\mid }}^{p-2}Du+a(x)\mathrm{\mid }Du{\mathrm{\mid }}^{q-2}Du)=0,$$ which is characterized by the fact that both ellipticity and growth switch between two different types of polynomial according to the position. We introduce the ${\mathcal{A}}_{H(\cdot )}$-harmonic functions of nonlinear potential theory, and then show that ${\mathcal{A}}_{H(\cdot )}$-harmonic functions coincide with the distributional and viscosity solutions, respectively. This implies that the distributional and viscosity solutions are exactly the same.