Partial differential equations
Ground state solution for a non-autonomous 1-Laplacian problem involving periodic potential in N
Comptes Rendus. Mathématique, Volume 360 (2022) no. G4, 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.

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DOI: 10.5802/crmath.276
Classification: 35J62, 35J75
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
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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, Volume 360 (2022) no. G4, pp. 297-304. doi : 10.5802/crmath.276. http://www.numdam.org/articles/10.5802/crmath.276/

[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 N , Bull. Braz. Math. Soc. (N.S.), Volume 51 (2020) no. 3, pp. 863-886 | DOI | MR | Zbl

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

[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 | DOI | MR | Zbl

[5] Che, Guofeng; Shi, Hongxia; Wang, Zewei 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] 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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

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

[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 | MR | Zbl

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

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

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