Nonhomogeneous nonlinear integral equations on bounded domains

Xing Yi; Xing Yi

doi:10.3934/math.20231132

AIMS Mathematics

2023, Volume 8, Issue 9: 22207-22224. doi: 10.3934/math.20231132

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Nonhomogeneous nonlinear integral equations on bounded domains

Xing Yi ^,

College of Mathematics and Statistics, Hunan Normal University, Hunan, China

Received: 23 May 2023 Revised: 29 June 2023 Accepted: 30 June 2023 Published: 12 July 2023
MSC : 45G05, 35A01, 35B44

This paper investigates the existence of positive solutions for a nonhomogeneous nonlinear integral equation of the form

$ \begin{equation} u^{p-1}(x) = \int_{\Omega} \frac{u(y)}{|x-y|^{n-\alpha}} d y+\int_{\Omega} \frac{f(y)}{|x-y|^{n-\alpha}} d y, \ x \in \bar{\Omega}\nonumber \end{equation} $

where $ \frac{2n}{n+\alpha}\leq p < 2, $ $ 1 < \alpha < n $, $ n > 2, $ $ \Omega $ is a bounded domain in $ \mathbb R^{n} $. We show that under suitable assumptions on $ f, $ the integral equation admits a positive solution in $ L^{\frac{2n}{n+\alpha}}\left(\Omega\right) $. Our method combines the Ekeland variational principle, a blow-up argument and a rescaling argument which allows us to overcome the difficulties arising from the lack of Brezis-Lieb lemma in $ L^{\frac{2n}{n+\alpha}}(\Omega) $.
- integral equation,
- Hardy-Littlewood-Sobolev inequality,
- blowing-up and rescaling argument,
- Ekeland variational principle
Citation: Xing Yi. Nonhomogeneous nonlinear integral equations on bounded domains[J]. AIMS Mathematics, 2023, 8(9): 22207-22224. doi: 10.3934/math.20231132

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Abstract

This paper investigates the existence of positive solutions for a nonhomogeneous nonlinear integral equation of the form

$ \begin{equation} u^{p-1}(x) = \int_{\Omega} \frac{u(y)}{|x-y|^{n-\alpha}} d y+\int_{\Omega} \frac{f(y)}{|x-y|^{n-\alpha}} d y, \ x \in \bar{\Omega}\nonumber \end{equation} $

where $ \frac{2n}{n+\alpha}\leq p < 2, $ $ 1 < \alpha < n $, $ n > 2, $ $ \Omega $ is a bounded domain in $ \mathbb R^{n} $. We show that under suitable assumptions on $ f, $ the integral equation admits a positive solution in $ L^{\frac{2n}{n+\alpha}}\left(\Omega\right) $. Our method combines the Ekeland variational principle, a blow-up argument and a rescaling argument which allows us to overcome the difficulties arising from the lack of Brezis-Lieb lemma in $ L^{\frac{2n}{n+\alpha}}(\Omega) $.

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