Anisotropic Choquard problems with Stein–Weiss potential: nonlinear patterns and stationary waves

Youpei Zhang; Xianhua Tang; Vicenţiu Rădulescu

doi:10.5802/crmath.253

Équations aux dérivées partielles

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.

Résumé

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.

Reçu le : 2021-06-29
Accepté le : 2021-08-03
Publié le : 2021-10-06

DOI : 10.5802/crmath.253

Classification : 35A23, 47J20, 58E05, 58E35

Affiliations des auteurs :

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

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Youpei Zhang and Xianhua Tang and Vicen\c{t}iu R\u{a}dulescu},
     title = {Anisotropic {Choquard} problems with {Stein{\textendash}Weiss} potential: nonlinear patterns and stationary waves},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {959--968},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {359},
     number = {8},
     year = {2021},
     doi = {10.5802/crmath.253},
     language = {en},
}

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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/

Bibliographie
Cité par

[1] Claudianor O. Alves Existence of radial solutions for a class of $p (x)$ -Laplacian equations with critical growth, Differ. Integral Equ., Volume 23 (2010) no. 1-2, pp. 113-123 | MR | Zbl

[2] Claudianor O. Alves; Leandro S. Tavares A Hardy–Littlewood–Sobolev-type inequality for variable exponents and applications to quasilinear Choquard equations involving variable exponent, Mediterr. J. Math., Volume 16 (2019) no. 2, 55, 27 pages | MR | Zbl

[3] Antonio Ambrosetti; Paul H. Rabinowitz Dual variational methods in critical point theory and applications, J. Funct. Anal., Volume 14 (1973), pp. 349-381 | DOI | MR | Zbl

[4] Luis Caffarelli; Robert Kohn; Louis Nirenberg First order interpolation inequalities with weights, Compos. Math., Volume 53 (1984) no. 3, pp. 259-275 | Numdam | MR | Zbl

[5] Xianling Fan; Jishen Shen; Dun Zhao Sobolev embedding theorems for spaces $W^{k, p (x)} (Ω)$ , J. Math. Anal. Appl., Volume 262 (2001) no. 2, pp. 749-760 | Zbl

[6] Godfrey H. Hardy; John E. Littlewood Some properties of fractional integrals. I, Math. Z., Volume 27 (1928) no. 1, pp. 565-606 | DOI | MR | Zbl

[7] In Hyoun Kim; Yun-Ho Kim Mountain pass type solutions and positivity of the infimum eigenvalue for quasilinear elliptic equations with variable exponents, Manuscr. Math., Volume 147 (2015) no. 1-2, pp. 169-191 | MR | Zbl

[8] Richard S. Palais; Steve Smale A generalized Morse theory, Bull. Am. Math. Soc., Volume 70 (1964), pp. 165-171 | DOI | MR | Zbl

[9] Patrizia Pucci; Vicenţiu D. Rădulescu The impact of the mountain pass theory in nonlinear analysis: a mathematical survey, Boll. Unione Mat. Ital. (9), Volume 3 (2010) no. 3, pp. 543-584 | MR | Zbl

[10] Boris Rubin One-dimensional representation, inversion and certain properties of Riesz potentials of radial functions, Mat. Zametki, Volume 34 (1983), pp. 521-533 translation in Math. Notes 34 (1983), p. 751-757 | MR | Zbl

[11] Sergeĭ Sobolev On a theorem of functional analysis, Mat. Sb., N. Ser., Volume 4 (1938), pp. 471-479 translation in Amer. Math. Soc. Transl. Ser. 2 34 (1963), p. 39-68

[12] Elias M. Stein; Guido Weiss Fractional integrals on $n$ -dimensional Euclidean space, J. Math. Mech., Volume 7 (1958), pp. 503-514 | MR | Zbl

[13] Michel Willem Functional Analysis. Fundamentals and Applications, Cornerstones, Birkhäuser, 2013 | MR | Zbl

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