Ground state solution for a non-autonomous 1-Laplacian problem involving periodic potential in $\protect \mathbb{R}^N$

Wang, Shi-Ying; Chen, Peng; Li, Lin

doi:10.5802/crmath.276

Geodesic

Parcourir par

Équations aux dérivées partielles

Ground state solution for a non-autonomous 1-Laplacian problem involving periodic potential in

ℝ^{N}

Wang, Shi-Ying ¹ ; Chen, Peng ¹ ; Li, Lin 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

Comptes Rendus. Mathématique, Tome 360 (2022) no. G4, pp. 297-304.

Voir la notice de l'article provenant de la source Numdam

Résumé

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

- Δ_{1} u + V (x) \frac{u}{| u |} = f (x, u), x \in ℝ^{N}, u > 0, u \in B V (ℝ^{N}),

where $Δ_{1} u = div (\frac{D u}{| D u |})$ , $V$ 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.

Reçu le : 2021-09-07
Accepté le : 2021-09-26
Accepté après révision le : 2021-10-15
Publié le : 2022-04-26

DOI : 10.5802/crmath.276

Classification : 35J62, 35J75

Affiliations des auteurs :

Wang, Shi-Ying ¹ ; Chen, Peng ¹ ; Li, Lin ²

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

@article{CRMATH_2022__360_G4_297_0,
     author = {Wang, Shi-Ying and Chen, Peng and Li, Lin},
     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},
     number = {G4},
     year = {2022},
     doi = {10.5802/crmath.276},
     language = {en},
     url = {https://geodesic-test.mathdoc.fr/articles/10.5802/crmath.276/}
}

TY  - JOUR
AU  - Wang, Shi-Ying
AU  - Chen, Peng
AU  - Li, Lin
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
IS  - G4
PB  - Académie des sciences, Paris
UR  - https://geodesic-test.mathdoc.fr/articles/10.5802/crmath.276/
DO  - 10.5802/crmath.276
LA  - en
ID  - CRMATH_2022__360_G4_297_0
ER  -

%0 Journal Article
%A Wang, Shi-Ying
%A Chen, Peng
%A Li, Lin
%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
%N G4
%I Académie des sciences, Paris
%U https://geodesic-test.mathdoc.fr/articles/10.5802/crmath.276/
%R 10.5802/crmath.276
%G en
%F CRMATH_2022__360_G4_297_0

Wang, Shi-Ying; Chen, Peng; Li, Lin. Ground state solution for a non-autonomous 1-Laplacian problem involving periodic potential in $\protect \mathbb{R}^N$. Comptes Rendus. Mathématique, Tome 360 (2022) no. G4, pp. 297-304. doi : 10.5802/crmath.276. https://geodesic-test.mathdoc.fr/articles/10.5802/crmath.276/

Bibliographie
Cité par

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

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

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

[4] Attouch, Hedy; Buttazzo, Giuseppe; Michaille, Gérard 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 | Zbl | MR | DOI

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

[6] Figueiredo, Giovany M.; Pimenta, Marcos T. O. 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 | Zbl | MR | DOI

[7] Figueiredo, Giovany M.; Pimenta, Marcos T. O. 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 | Zbl | MR | DOI

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

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

[10] Lions, Pierre-Louis 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 | Zbl | MR

[11] Ortiz Chata, Juan C.; Pimenta, Marcos T. O. 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 | Zbl | MR | DOI

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

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

Cité par Sources :