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

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