  The purpose of this paper is to consider the minimization problem of the
following nonlocal interaction functional \begin{equation*}
E[\rho]=\frac{1}{2}\int_{\mathbb{R}^N}
\int_{\mathbb{R}^N}K(x-y)\rho(x)\rho(y)dxdy+\int_{\mathbb{R}^N}F(x)\rho(x)dx.
\end{equation*} The kernel $K(x)=\frac{1}{q}|x|^q-\frac{1}{p}|x|^p$ is an
endogenous potential, where $q>p>-N$. The exogenous potential $F$ is a
nonnegative continuous function and satisfies $F(x)\to +\infty$ as $|x|\to
+\infty$. The existence of minimizers are established based on the
concentration compactness principle. Especially, for $F(x)=\beta|x|^2(\beta
>0)$ and $K(x)=\frac{1}{2}|x|^2-\frac{1}{2-N}|x|^{2-N}$($N>2$), the global
minimizer is given explicitly by the method of calculus of variation.
