Regularity results for solutions of mixed local and nonlocal elliptic equations

We consider the mixed local-nonlocal semi-linear elliptic equations driven by the superposition of Brownian and Lévy processes \begin{equation*} \left{ \begin{array}{ll} - Δu + (-Δ)^s u = g(x,u) & \hbox{in $\mathrm{\Omega }$,} u=0 & \hbox{in ${\mathbb{R}}^n\mathrm{\backslash }\mathrm{\Omega }$.} \ \end{array} \right. \end{equation*} Under mild assumptions on the nonlinear term $g$, we show the $L^{\mathrm{\infty }}$ boundedness of any weak solution (either not changing sign or sign-changing) by the Moser iteration method. Moreover, when $s\in (0,\frac{1}{2}]$, we obtain that the solution is unique and actually belongs to $C^{1,\alpha }(\bar{\mathrm{\Omega }})$ for any $\alpha \in (0,1)$.