This work focuses on the nonhomogeneous nonlocal double phase problem \begin{ align*} L_au(x)=f(x,u,D_s^p u, D_{ a,t}^q u) \text{ in } \Omega, \end{ align*} where $\Omega\subset\mathbb{ R}^N$ is a bounded domain with Lipschitz boundary, $0<s,t<1<p\leq q<\infty$ with $tq\leq sp$ and the operator $L_a$ is defined as \begin{ align*} L_a u(x)&=2\operatorname{ P.V.}\int_{ \mathbb{ R}^N}|u(x)-u(y)|^{ p-2}(u(x)-u(y))K_{ s,p}(x,y) &\ \ \ +2\operatorname{ P.V.}\int_{ \mathbb{ R}^N}a(x,y)|u(x)-u(y)|^{ q-2}(u(x)-u(y))K_{ t,q}(x,y)dy. \end{ align*} We establish the equivalence between weak and viscosity solutions under boundedness and continuity assumptions. In addition, the local boundedness of weak solutions in some special cases on $f$ is also obtained using the notion of De Giorgi classes.
Categories: math.AP