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Comptes Rendus. Mathématique
Partial differential equations
Anisotropic Choquard problems with Stein–Weiss potential: nonlinear patterns and stationary waves
Youpei Zhang 1, 2; Xianhua Tang 2; Vicenţiu Rădulescu 1, 3
1. Department of Mathematics, University of Craiova, Craiova 200585, Romania
2. School of Mathematics and Statistics, Central South University, Changsha, Hunan 410083, China
3. Faculty of Applied Mathematics, AGH University of Science and Technology, al. Mickiewicza 30, 30-059 Kraków, Poland
Comptes Rendus. Mathématique, Volume 359 (2021) no. 8, pp. 959-968.

Weighted inequality theory for fractional integrals is a relatively less known branch of calculus that offers remarkable opportunities to simulate interdisciplinary processes. Basic weighted inequalities are often associated to Hardy, Littlewood and Sobolev [6, 11], Caffarelli, Kohn and Nirenberg [4], respectively to Stein and Weiss [12]. A key attempt in the present paper is to prove a Stein–Weiss inequality with lack of symmetry and variable exponents. We quantify the defect of symmetry of the potential by considering the gap between the minimum and the maximum of the variable exponent. We conclude our work with a section dealing with the existence of stationary waves for a class of nonlocal problems with Choquard nonlinearity and anisotropic Stein–Weiss potential.

Received:
Accepted:
Published online:
DOI: https://doi.org/10.5802/crmath.253
Classification: 35A23,  47J20,  58E05,  58E35
Youpei Zhang 1, 2; Xianhua Tang 2; Vicenţiu Rădulescu 1, 3

1. Department of Mathematics, University of Craiova, Craiova 200585, Romania
2. School of Mathematics and Statistics, Central South University, Changsha, Hunan 410083, China
3. Faculty of Applied Mathematics, AGH University of Science and Technology, al. Mickiewicza 30, 30-059 Kraków, Poland 
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     title = {Anisotropic {Choquard} problems with {Stein{\textendash}Weiss} potential: nonlinear patterns and stationary waves},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {959--968},
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Youpei Zhang; Xianhua Tang; Vicenţiu Rădulescu. Anisotropic Choquard problems with Stein–Weiss potential: nonlinear patterns and stationary waves. Comptes Rendus. Mathématique, Volume 359 (2021) no. 8, pp. 959-968. doi : 10.5802/crmath.253. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.253/

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