An integral system of Matukuma type with negative exponents

Changhong Li; Changhong Li

doi:10.3934/nhm.2025025

Networks and Heterogeneous Media

2025, Volume 20, Issue 2: 566-589. doi: 10.3934/nhm.2025025

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An integral system of Matukuma type with negative exponents

Changhong Li ^,

School of Mathematics and Statistics, University College Dublin, Belfield, Dublin 4, Ireland

Received: 11 March 2025 Revised: 09 May 2025 Accepted: 16 May 2025 Published: 28 May 2025
45G15, 45M20, 47H10, 47J05, 35J48

We study positive solutions to the integral system
$ \begin{cases} u(x) = \int\limits_{ \mathbb R^N}\frac{K(y)v^{-p}(y)}{ |x-y|^{N-\alpha}} dy\quad\text{ for all }x\in \mathbb R^N, \\ v(x) = \int\limits_{ \mathbb R^N}\frac{L(y)u^{-q}(y)}{ |x-y|^{N-\beta}} dy\quad\text{ for all }x\in \mathbb R^N, \end{cases} $
where $ p, q > 0 $, $ \alpha, \beta\in (0, N) $ and $ K, L: \mathbb R^N\to (0, \infty) $ are continuous functions which satisfy $ C_1(1+|x|)^{-\gamma}\leq K(x), L(x)\leq C_2(1+|x|)^{-\gamma} $ in $ \mathbb R^N $, for some $ \gamma > 0 $ and constants $ C_2 > C_1 > 0 $. We discuss the existence, nonexistence, and uniqueness of positive solutions to the above system with respect to $ \alpha, \beta, p, q, $ and $ \gamma $. We also classify the finite and infinite total mass solutions of the system.
- general weighted integral system,
- negative exponent,
- Matukuma type equation,
- Riesz potentials,
- existence and uniqueness of positive solution
Citation: Changhong Li. An integral system of Matukuma type with negative exponents[J]. Networks and Heterogeneous Media, 2025, 20(2): 566-589. doi: 10.3934/nhm.2025025

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Abstract

We study positive solutions to the integral system

$ \begin{cases} u(x) = \int\limits_{ \mathbb R^N}\frac{K(y)v^{-p}(y)}{ |x-y|^{N-\alpha}} dy\quad\text{ for all }x\in \mathbb R^N, \\ v(x) = \int\limits_{ \mathbb R^N}\frac{L(y)u^{-q}(y)}{ |x-y|^{N-\beta}} dy\quad\text{ for all }x\in \mathbb R^N, \end{cases} $

where $ p, q > 0 $, $ \alpha, \beta\in (0, N) $ and $ K, L: \mathbb R^N\to (0, \infty) $ are continuous functions which satisfy $ C_1(1+|x|)^{-\gamma}\leq K(x), L(x)\leq C_2(1+|x|)^{-\gamma} $ in $ \mathbb R^N $, for some $ \gamma > 0 $ and constants $ C_2 > C_1 > 0 $. We discuss the existence, nonexistence, and uniqueness of positive solutions to the above system with respect to $ \alpha, \beta, p, q, $ and $ \gamma $. We also classify the finite and infinite total mass solutions of the system.

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