In this paper, we study the elementary solution of the operator $circledast_B^k$ which is defined by $$circledast_B^k=left[left(B_{x_1}+B_{x_2}+cdots+B_{x_p} ight)^3 +left(B_{x_{p+1}}+cdots+B_{x_{p+q}} ight)^3 ight]^k,$$ where $p+q=n$ is the dimension of $mathbb{R}^+_n={(x=x_1,x_2,dots,x_n):x_1>0,x_2>0,dots,x_n>0}$, $B_{x_i}=frac{partial^2}{partial x_i^2}+ frac{2v_i}{x_i}frac{partial}{partial x_i}$, $2v_i=2alpha_i+1$, $alpha_i>-frac{1}{2}$, $x_i>0$, $i=1,2,dots,n$ and $k$ is a positive integer. After that, we apply such an elementary solution to solve the equation $circledast_B^ku(x)=f(x)$, where $f$ is a generalized function and $u$ is an unknown function.
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