| In this paper, we consider the Hopf lemma of the following mixed local and nonlocal weighted semilinear elliptic equations
\begin{align*}
\left \{\begin{array}{rl}
- \text{div}(|x|^{-2\alpha}\nabla u)+(-\Delta)_\alpha^s u =0, &x\in U,\~u(\bar{x})=-u(x),&x\in H,\~u(x)=0,&x\in \mathbb{R}^N\setminus U,
\end{array}
\right.
\end{align*}
where $H\subset\mathbb{R}^N$ with $0\in H$ is an open and affine half space, $U\subset H$ is an open and bounded set, $s\in(0,1), \alpha\in[ 0,\frac{N-2s}{2})$, $(-\Delta)_\alpha^s$ is weighted fractional Laplacian with a weighted function. |
