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

Youpei Zhang; Xianhua Tang; Vicenţiu Rădulescu

doi:10.5802/crmath.253

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.

Abstract

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: 2021-06-29
Accepted: 2021-08-03
Published online: 2021-10-06

DOI: 10.5802/crmath.253

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

Author's affiliations:

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

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

@article{CRMATH_2021__359_8_959_0,
     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},
}

TY  - JOUR
AU  - Youpei Zhang
AU  - Xianhua Tang
AU  - Vicenţiu Rădulescu
TI  - Anisotropic Choquard problems with Stein–Weiss potential: nonlinear patterns and stationary waves
JO  - Comptes Rendus. Mathématique
PY  - 2021
SP  - 959
EP  - 968
VL  - 359
IS  - 8
PB  - Académie des sciences, Paris
DO  - 10.5802/crmath.253
LA  - en
ID  - CRMATH_2021__359_8_959_0
ER  -

%0 Journal Article
%A Youpei Zhang
%A Xianhua Tang
%A Vicenţiu Rădulescu
%T Anisotropic Choquard problems with Stein–Weiss potential: nonlinear patterns and stationary waves
%J Comptes Rendus. Mathématique
%D 2021
%P 959-968
%V 359
%N 8
%I Académie des sciences, Paris
%R 10.5802/crmath.253
%G en
%F CRMATH_2021__359_8_959_0

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/

References
Cited by

[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

Cited by Sources:

Comments - Policy