In this paper, we consider the minimization of interaction functional$$\begin{aligned} E[\rho ]= & {} \int _{{\mathbb {R}}^N}\frac{1}{a-1}\rho ^a(x)\mathrm{d}x+\frac{1}{2}\int _{{\mathbb {R}}^N} \int _{{\mathbb {R}}^N}K(x-y)\rho (x)\rho (y)\mathrm{d}x\mathrm{d}y\nonumber \\&+\int _{{\mathbb {R}}^N}F(x)\rho (x)\mathrm{d}x, \end{aligned}$$(0.1)where \(a>1\), the term \(\frac{1}{a-1}\rho ^a\) is homogeneous nonlinear diffusion, and the kernel \(K(x)=\frac{1}{q}|x|^q-\frac{1}{p}|x|^p\) is an endogenous potential with \(q>p>-N\). The exogenous potential F is a nonnegative function satisfying some extra conditions. We analyze the existence of minimizers for the functional (0.1). Furthermore, we show the explicit of global minimizer of (0.1) when \(q=2\), \(p=2-N\) and F is a power function.

中文翻译：

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具有非线性扩散和外生势的相互作用函数

在本文中，我们考虑了交互函数的最小化$$ \ begin {aligned} E [\ rho] =＆{} \ int _ {{\\ mathbb {R}} ^ N} \ frac {1} {a-1 } \ rho ^ a（x）\ mathrm {d} x + \ frac {1} {2} \ int _ {{\ mathbb {R}} ^ N} \ int _ {{\ mathbb {R}} ^ N} K（xy）\ rho（x）\ rho（y）\ mathrm {d} x \ mathrm {d} y \ nonumber \\＆+ \ int _ {{\ mathbb {R}} ^ N} F（x） \ rho（x）\ mathrm {d} x，\ end {aligned} $$（0.1）其中\（a> 1 \），术语\（\ frac {1} {a-1} \ rho ^ a \ ）是齐次的非线性扩散，并且核\（K（x）= \ frac {1} {q} | x | ^ q- \ frac {1} {p} | x | ^ p \）是具有\（q> p> -N \）。外生势F是满足某些额外条件的非负函数。我们分析功能（0.1）的最小化器的存在。此外，当\（q = 2 \），\（p = 2-N \）和F是幂函数时，我们显示（0.1）的全局最小化器的显式性。