Comptes Rendus
Partial differential equations
Ground state solution for a non-autonomous 1-Laplacian problem involving periodic potential in N
Comptes Rendus. Mathématique, Volume 360 (2022), pp. 297-304.

In this paper, we consider the following 1-Laplacian problem

-Δ 1 u+V(x)u |u|=f(x,u),x N ,u>0,uBV N ,

where Δ 1 u=div(Du |Du|), V is a periodic potential and f is periodic and verifies the super-primary condition at infinity. By the Nehari type manifold method, the concentration compactness principle and some analysis techniques, we show the 1-Laplacian equation has a ground state solution.

Received:
Accepted:
Revised after acceptance:
Published online:
DOI: 10.5802/crmath.276
Classification: 35J62, 35J75
Shi-Ying Wang 1; Peng Chen 1; Lin Li 2

1 School of Science, China Three Gorges University, Hubei 443002, China
2 School of Mathematics and Statistics & Chongqing Key Laboratory of Economic and Social Application Statistics, Chongqing Technology and Business University, Chongqing 400067, China
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
@article{CRMATH_2022__360_G4_297_0,
     author = {Shi-Ying Wang and Peng Chen and Lin Li},
     title = {Ground state solution for a non-autonomous {1-Laplacian} problem involving periodic potential in $\protect \mathbb{R}^N$},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {297--304},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {360},
     year = {2022},
     doi = {10.5802/crmath.276},
     language = {en},
}
TY  - JOUR
AU  - Shi-Ying Wang
AU  - Peng Chen
AU  - Lin Li
TI  - Ground state solution for a non-autonomous 1-Laplacian problem involving periodic potential in $\protect \mathbb{R}^N$
JO  - Comptes Rendus. Mathématique
PY  - 2022
SP  - 297
EP  - 304
VL  - 360
PB  - Académie des sciences, Paris
DO  - 10.5802/crmath.276
LA  - en
ID  - CRMATH_2022__360_G4_297_0
ER  - 
%0 Journal Article
%A Shi-Ying Wang
%A Peng Chen
%A Lin Li
%T Ground state solution for a non-autonomous 1-Laplacian problem involving periodic potential in $\protect \mathbb{R}^N$
%J Comptes Rendus. Mathématique
%D 2022
%P 297-304
%V 360
%I Académie des sciences, Paris
%R 10.5802/crmath.276
%G en
%F CRMATH_2022__360_G4_297_0
Shi-Ying Wang; Peng Chen; Lin Li. Ground state solution for a non-autonomous 1-Laplacian problem involving periodic potential in $\protect \mathbb{R}^N$. Comptes Rendus. Mathématique, Volume 360 (2022), pp. 297-304. doi : 10.5802/crmath.276. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.276/

[1] Claudianor O. Alves A Berestycki–Lions type result for a class of problems involving the 1-Laplacian operator, Commun. Contemp. Math. (2021), 2150022 | DOI

[2] Claudianor O. Alves; Giovany M. Figueiredo; Marcos T. O. Pimenta Existence and profile of ground-state solutions to a 1-Laplacian problem in N , Bull. Braz. Math. Soc. (N.S.), Volume 51 (2020) no. 3, pp. 863-886 | DOI | MR | Zbl

[3] Gabriele Anzellotti The Euler equation for functionals with linear growth, Trans. Am. Math. Soc., Volume 290 (1985) no. 2, pp. 483-501 | DOI | MR | Zbl

[4] Hedy Attouch; Giuseppe Buttazzo; Gérard Michaille Variational analysis in Sobolev and BV spaces. Applications to PDEs and optimization, MOS-SIAM Series on Optimization, 17, Society for Industrial and Applied Mathematics; Mathematical Optimization Society, Philadelphia, PA, 2014 | DOI | MR | Zbl

[5] Guofeng Che; Hongxia Shi; Zewei Wang Existence and concentration of positive ground states for a 1-Laplacian problem in N , Appl. Math. Lett., Volume 100 (2020), 106045 | DOI | MR | Zbl

[6] Giovany M. Figueiredo; Marcos T. O. Pimenta Existence of bounded variation solutions for a 1-Laplacian problem with vanishing potentials, J. Math. Anal. Appl., Volume 459 (2018) no. 2, pp. 861-878 | DOI | MR | Zbl

[7] Giovany M. Figueiredo; Marcos T. O. Pimenta Nehari method for locally Lipschitz functionals with examples in problems in the space of bounded variation functions, NoDEA, Nonlinear Differ. Equ. Appl., Volume 25 (2018) no. 5, 47 | DOI | MR | Zbl

[8] Giovany M. Figueiredo; Marcos T. O. Pimenta Strauss’ and Lions’ type results in BV( N ) with an application to an 1-Laplacian problem, Milan J. Math., Volume 86 (2018) no. 1, pp. 15-30 | DOI | MR | Zbl

[9] Yongqing Li; Zhi-Qiang Wang; Jing Zeng Ground states of nonlinear Schrödinger equations with potentials, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 23 (2006) no. 6, pp. 829-837 | DOI | Numdam | MR | Zbl

[10] Pierre-Louis Lions The concentration-compactness principle in the calculus of variations. The locally compact case. I, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 1 (1984) no. 2, pp. 109-145 | DOI | MR | Zbl

[11] Juan C. Ortiz Chata; Marcos T. O. Pimenta A Berestycki–Lions’ type result to a quasilinear elliptic problem involving the 1-Laplacian operator, J. Math. Anal. Appl., Volume 500 (2021) no. 1, 125074 | DOI | MR | Zbl

[12] Leonid I. Rudin; Stanley Osher; Emad Fatemi Nonlinear total variation based noise removal algorithms, Physica D, Volume 60 (1992) no. 1-4, pp. 259-268 | DOI | MR | Zbl

[13] Fen Zhou; Zifei Shen Existence of a radial solution to a 1-Laplacian problem in N , Appl. Math. Lett., Volume 118 (2021), 107138 | DOI | MR | Zbl

Cited by Sources:

Comments - Policy


Articles of potential interest

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

Youpei Zhang; Xianhua Tang; Vicenţiu Rădulescu

C. R. Math (2021)


A distributional approach to fractional Sobolev spaces and fractional variation: asymptotics II

Elia Bruè; Mattia Calzi; Giovanni E. Comi; ...

C. R. Math (2022)


Stochastic control with state constraints via the Fokker–Planck equation. Application to renewable energy plants with batteries

Alfredo Bermúdez; Iago Padín

C. R. Méca (2023)