Existence of solutions to elliptic equation with mixed local and nonlocal operators

In this paper, making use of a new non-smooth variational approach established by Moameni^{[13,14,15,16]}, we establish the existence of solutions to the following mixed local and nonlocal elliptic problem

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

where $\mathrm{\Omega }\subset {\mathbb{R}}^N$ is a bounded smooth domain, $(-\mathrm{\Delta }{)}^s$ is the restricted fractional Laplacian, $\mu >0$, $0<s<1$, $N>2s$, $g$ satisfies some growth condition and $b(x)\in L^m(\mathrm{\Omega })$ for $m\ge 2$. The interesting feature of our work is that we show that the nonlocal operator has an important influence in the existence of solutions to the above equation since $g$ has new growth condition.