Regularity results for solutions of mixed local and nonlocal elliptic equations

Abstract

We consider the mixed local-nonlocal semi-linear elliptic equations driven by the superposition of Brownian and Lévy processes $$\begin{aligned} \left\{ \begin{array}{ll} - \Delta u + (-\Delta )^s u = g(x,u) &{} \text {in } \Omega , \\ u=0 &{} \text {in } \mathbb {R}^n\backslash \Omega . \\ \end{array} \right. \end{aligned}$$ Under mild assumptions on the nonlinear term g, we show the $$L^\infty $$ 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 $$C^{1,\alpha }(\overline{\Omega })$$ for any $$\alpha \in (0,1)$$ .