# Convergence of ground state solutions for nonlinear Schrödinger equations on graphs

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## Abstract

We consider the nonlinear Schrödinger equation -Δ*u* + (λ*a*(*x*) + 1)*u* = |*u*|^{p-1}*u* on a locally finite graph *G* = (*V*,*E*). We prove via the Nehari method that if *a*(*x*) satisfies certain assumptions, for any λ > 1, the equation admits a ground state solution *u*λ. Moreover, as λ → 1, the solution *u*_{λ} converges to a solution of the Dirichlet problem -Δ*u*+*u* = |*u*|^{p-1}*u* which is defined on the potential well Ω. We also provide a numerical experiment which solves the equation on a finite graph to illustrate our results.

## Keywords

Schrödinger equation locally finite graph ground state potential well## MSC(2010)

35A15 35Q55 58E30## Preview

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## Notes

### Acknowledgements

The first author was supported by the Funding of Beijing Philosophy and Social Science (Grant No. 15JGC153) and the Ministry of Education Project of Humanities and Social Sciences (Grant No. 16YJCZH148). The second author was supported by the Fundamental Research Funds for the Central Universities. Both of the authors were supported by the Ministry of Education Project of Key Research Institute of Humanities and Social Sciences at Universities (Grant No. 16JJD790060). The authors thank members of Data Lighthouse for their helpful conversations and valuable suggestions. The authors are also thankful for the referees′ detailed and useful comments.

## References

- 1.Alves C O, Souto M A S. On the existence and concentration behavior of ground state solutions for a class of problems with critical growth. Commun Pure Appl Anal, 2002, 1: 417–431MathSciNetCrossRefzbMATHGoogle Scholar
- 2.Bartsch T, Wang Z Q. Multiple positive solutions for a nonlinear Schrödinger equation. Z Angew Math Phys, 2000, 51: 366–384MathSciNetCrossRefzbMATHGoogle Scholar
- 3.Bauer F, Horn P, Lin Y, et al. Li-Yau inequality on graphs. J Differential Geom, 2015, 99: 359–405MathSciNetCrossRefzbMATHGoogle Scholar
- 4.Brézis H, Nirenberg L. Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents. Comm Pure Appl Math, 1983, 36: 437–477MathSciNetCrossRefzbMATHGoogle Scholar
- 5.Cao D M. Nontrivial solution of semilinear equations with critical exponent in R
^{2}. Comm Partial Differential Equations, 1992, 17: 407–435MathSciNetCrossRefGoogle Scholar - 6.Chakik A E, Elmoataz A, Desquesnes X. Mean curvature flow on graphs for image and manifold restoration and enhancement. Signal Processing, 2014, 105: 449–463CrossRefGoogle Scholar
- 7.Chung Y S, Lee Y S, Chung S Y. Extinction and positivity of the solutions of the heat equations with absorption on networks. J Math Anal Appl, 2011, 380: 642–652MathSciNetCrossRefzbMATHGoogle Scholar
- 8.Clapp M, Ding Y H. Positive solutions of a Schrödinger equation with critical nonlinearity. Z Angew Math Phys, 2004, 55: 592–605MathSciNetCrossRefzbMATHGoogle Scholar
- 9.Curtis E, Morrow J. Determining the resistors in a network. SIAM J Appl Math, 1990, 50: 918–930MathSciNetCrossRefzbMATHGoogle Scholar
- 10.Desquesnes X, Elmoataz A, Lézoray O. Eikonal equation adaptation on weighted graphs: Fast geometric diffusion process for local and non-local image and data processing. J Math Imaging Vision, 2013, 46: 238–257MathSciNetCrossRefzbMATHGoogle Scholar
- 11.Ding Y H, Tanaka K. Multiplicity of positive solutions of a nonlinear Schrödinger equation. Manuscripta Math, 2003, 112: 109–135MathSciNetCrossRefzbMATHGoogle Scholar
- 12.Elmoataz A, Desquesnes X, Lézoray O. Non-local morphological PDEs and
*p*-Laplacian equation on graphs with applications in image processing and machine learning. IEEE J Sel Top Signal Process, 2012, 6: 764–779CrossRefzbMATHGoogle Scholar - 13.Elmoataz A, Lézoray O, Bougleux S. Nonlocal discrete regularization on weighted graphs: A framework for image and manifold processing. IEEE Trans Image Process, 2008, 17: 1047–1060MathSciNetCrossRefzbMATHGoogle Scholar
- 14.Elmoataz A, Lozes F, Toutain M. Nonlocal PDEs on graphs: From Tug-of-War games to unified interpolation on images and point clouds. J Math Imaging Vision, 2017, 57: 381–401MathSciNetCrossRefzbMATHGoogle Scholar
- 15.Elmoataz A, Toutain M, Tenbrinck D. On the
*p*-Laplacian and ∞-Laplacian on graphs with applications in image and data processing. SIAM J Imaging Sci, 2015, 8: 2412–2451MathSciNetCrossRefzbMATHGoogle Scholar - 16.Grigor’yan A, Lin Y, Yang Y Y. Yamabe type equations on graphs. J Differential Equations, 2016, 261: 4924–4943MathSciNetCrossRefzbMATHGoogle Scholar
- 17.Grigor’yan A, Lin Y, Yang Y Y. Kazdan-Warner equation on graph. Calc Var Partial Differential Equations, 2016, 55: 92MathSciNetCrossRefzbMATHGoogle Scholar
- 18.Grigor’yan A, Lin Y, Yang Y Y. Existence of positive solutions to some nonlinear equations on locally finite graphs. Sci China Math, 2017, 60: 1311–1324MathSciNetCrossRefzbMATHGoogle Scholar
- 19.He X M, Zou W M. Existence and concentration of ground states for Schrödinger-Poisson equations with critical growth. J Math Phys, 2012, 53: 1–19CrossRefzbMATHGoogle Scholar
- 20.Horn P, Lin Y, Liu S, et al. Volume doubling, Poincare inequality and Gaussian heat kernel estimate for non-negatively curved graphs. J Reine Angew Math, 2017, doi:10.1515/crelle-2017–0038Google Scholar
- 21.Huang X P. On uniqueness class for a heat equation on graphs. J Math Anal Appl, 2012, 393: 377–388MathSciNetCrossRefzbMATHGoogle Scholar
- 22.Li Y Q, Wang Z Q, Zeng J. Ground states of nonlinear Schrödinger equations with potentials. Ann Inst H Poincaré Anal Non Linéaire, 2006, 23: 829–837MathSciNetCrossRefzbMATHGoogle Scholar
- 23.Lin Y, Wu Y T. On-diagonal lower estimate of heat kernels on graphs. J Math Anal Appl, 2017, 456: 1040–1048MathSciNetCrossRefzbMATHGoogle Scholar
- 24.Lin Y, Wu Y T. The existence and nonexistence of global solutions for a semilinear heat equation on graphs. Calc Var Partial Differential Equations, 2017, 56: 102MathSciNetCrossRefzbMATHGoogle Scholar
- 25.Liu W J, Chen K W, Yu J. Extinction and asymptotic behavior of solutions for the
*ω*-heat equation on graphs with source and interior absorption. J Math Anal Appl, 2016, 435: 112–132MathSciNetCrossRefzbMATHGoogle Scholar - 26.Nehari Z. On a class of nonlinear second-order differential equations. Trans Amer Math Soc, 1960, 95: 101–123MathSciNetCrossRefzbMATHGoogle Scholar
- 27.Rabinowitz P H. On a class of nonlinear Schrödinger equations. Z Angew Math Phys, 1992, 43: 270–291MathSciNetCrossRefzbMATHGoogle Scholar
- 28.Wang Z P, Zhou H S. Positive solutions for nonlinear Schrödinger equations with deepening potential well. J Eur Math Soc (JEMS), 2009, 11: 545–573MathSciNetCrossRefzbMATHGoogle Scholar
- 29.Wojciechowski R. Heat kernel and essential spectrum of infinite graphs. Indiana Univ Math J, 2009, 58: 1419–1441MathSciNetCrossRefzbMATHGoogle Scholar
- 30.Xin Q, Xu L, Mu C. Blow-up for the
*ω*-heat equation with Dirichlet boundary conditions and a reaction term on graphs. Appl Anal, 2014, 93: 1691–1701MathSciNetCrossRefzbMATHGoogle Scholar - 31.Yang Y Y. Existence of positive solutions to quasi-linear elliptic equations with exponential growth in the whole Euclidean space. J Funct Anal, 2012, 262: 1679–1704MathSciNetCrossRefzbMATHGoogle Scholar
- 32.Yang Y Y, Zhao L. A class of Adams-Fontana type inequalities and related functionals on manifolds. NoDEA Nonlinear Differential Equations Appl, 2010, 17: 119–135MathSciNetCrossRefzbMATHGoogle Scholar
- 33.Zhao L, Chang Y Y. Min-max level estimate for a singular quasilinear polyharmonic equation in R
^{2m}. J Differential Equations, 2013, 254: 2434–2464MathSciNetCrossRefzbMATHGoogle Scholar - 34.Zhao L, Zhang N. Existence of solutions for a higher order Kirchhoff type problem with exponetial critical growth. Nonlinear Anal, 2016, 132: 214–226MathSciNetCrossRefzbMATHGoogle Scholar