Regularity results of solutions to elliptic equations involving mixed local and nonlocal operators

In this paper, we study the summability of solutions to the following semilinear elliptic equations involving mixed local and nonlocal operators

$$\{\begin{array}{cc}-\mathrm{\Delta }u(x)+{(-\mathrm{\Delta })}^{s}u(x)=f(x),& x\in \mathrm{\Omega },\\ u(x)\ge 0,& x\in \mathrm{\Omega },\\ u(x)=0,& x\in {\mathbb{R}}^N\setminus \mathrm{\Omega },\end{array}$$

where $0<s<1$, $\mathrm{\Omega }\subset {\mathbb{R}}^N$ is a smooth bounded domain, $(-\mathrm{\Delta }{)}^s$ is the fractional Laplace operator, $f$ is a measurable function.