Abstract
This paper is mainly concerned with the following nonlinear p-Laplacian equation
on a locally finite graph G = (V, E) with more general nonlinear term, where Δp is the discrete p-Laplacian on graphs, p ≥ 2. Under some suitable conditions on f and a(x), we can prove that the equation admits a positive solution by the Mountain Pass theorem and a ground state solution uλ via the method of Nehari manifold, for any λ > 1. In addition, as λ → +∞, we prove that the solution uλ converge to a solution of the following Dirichlet problem
where Ω = {x ∈ V:a(x) = 0} is the potential well and ∂Ω denotes the the boundary of Ω.
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References
Adams, R. A.: Sobolev Spaces, Academic Press, 1975
Bartsch, T., Wang, Z. Q.: Multiple positive solutions for a nonlinear Schrödinger equation. Z. Angew. Math. Phys., 51, 366–384 (2000)
Brézis, H.: Nirenberg, L., Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents. Comm. Pure Appl. Math., 36, 437–477 (1983)
Cao, D. M.: Nontrivial solution of semilinear equations with critical exponent in ℝ2. Commun. Partial Differential Equations, 17, 407–435 (1992)
Chen, L., Coulhon, T., Hua, B.: Riesz transforms for bounded Laplacians on graphs. Math. Z., 294, 397–417 (2020)
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., 380, 642–652 (2011)
Clapp, M., Ding, Y. H.: Positive solutions of a Schrödinger equation with critical nonlinearity. Z. Angew. Math. Phys., 55, 592–605 (2004)
Ding, G.: Introduction to Banach Spaces (in Chinese), Science Press, 1997
Ding, Y. H., Tanaka, K.: Multiplicity of positive solutions of a nonlinear Schrödinger equation. Manuscripta Mathematica, 112, 109–135 (2003)
Ge, H. B., Jiang, W. F.: Kazdan-Warner equation on infinite graphs. J. Korean Math. Soc., 55, 1091–1101 (2018)
Ge, H. B., Jiang, W. F.: Yamabe equations on infinite graphs. J. Math. Anal. Appl., 460, 885–890 (2018)
Ge, H. B.: A p-th Yamabe equation on graph. Proceedings of the American Mathematical Society, 146(5), 2219–2224 (2018)
Grigor’yan, A., Lin, Y., Yang, Y. Y.: Kazdan-Warner equation on graph. Calc. Var. Partial Differential Equations, 55(4), Art. 92, 13 pp. (2016)
Grigor’yan, A., Lin, Y., Yang, Y. Y.: Yamabe type equations on graphs. J. Differential Equations, 261, 4924–4943 (2016)
Grigor’yan, A., Lin, Y., Yang, Y. Y.: Existence of positive solutions to some nonlinear equations on locally finite graphs. Sci. China Math., 60, 1311–1324 (2017)
Han, X., Shao, M., Zhao, L.: Existence and convergence of solutions for nonlinear biharmonic equations on graphs. Journal of Differential Equations, 268(7), 3936–3961 (2020)
He, X. M., Zou, W. M.: Existence and concentration of ground states for Schroödinger-Poisson equations with critical growth. Journal of Mathematical Physics, 53, 1–19 (2012)
Huang, X. P.: On uniqueness class for a heat equation on graphs. J. Math. Anal. Appl., 393, 377–388 (2012)
Keller, M., Schwarz, M.: The Kazdan-Warner equation on canonically compactifiable graphs. Calc. Var. Partial Differential Equations, 57(2) Art. 70, 18 pp. (2018)
Lê, A: Eigenvalue problems for the p-Laplacian. Nonlinear Analysis, 64, 1057–1099 (2006)
Li, Y. Q., Wang, Z. Q., Zeng, J.: Ground states of nonlinear Schröodinger equations with potentials. Ann. Inst. H. Poincaré Anal. Non Linéaire, 23, 829–837 (2006)
Lin, Y., Wu, Y. T.: The existence and nonexistence of global solutions for a semilinear heat equation on graphs. Calc. Var. Partial Differential Equations, 56(4), Art. 102, 22 pp. (2017)
Lin, Y., Wu, Y. T.: On-diagonal lower estimate of heat kernels on graphs. J. Math. Anal. Appl., 456, 1040–1048 (2017)
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., 435, 112–132 (2016)
Nehari, Z.: On a class of nonlinear second-order differential equations. Trans. AMS, 95, 101–123 (1960)
Rabinowitz, P. H.: On a class of nonlinear Schröodinger equations. Z. Angew. Math. Phys., 43, 270–291 (1992)
Shao, M., Mao, A.: Multiplicity of solutions to Schröodinger-Poisson system with concave-convex nonlinearities. Applied Mathematics Letters, 83, 212–218 (2018)
Willem, M.: Minimax Theorems, Birkhaäuser, Boston, 1996
Wojciechowski, R. K.: Heat kernel and essential spectrum of infinite graphs. Indiana Univ. Math. J., 58, 1419–1441 (2009)
Xin, Q., Xu, L., Mu, C.: Blow-up for the ω-heat equation with Dirichelet boundary conditions and a reaction term on graphs. Appl. Anal., 93, 1691–1701 (2014)
Zhang, N., Zhao, L.: Convergence of ground state solutions for nonlinear Schrödinger equations on graphs. Sci. China Math., 61(8), 1481–1494 (2018)
Zhao, L., Chang, Y. Y., Min-max level estimate for a singular quasilinear polyharmonic equation in ℝ2m. J. Differential Equations, 254, 2434–2464 (2013)
Zhao, L., Zhang, N.: Existence of solutions for a higher order Kirchhoff type problem with exponetial critical growth. Nonlinear Anal., 132, 214–226 (2016)
Acknowledgements
The authors are grateful to Prof. Zhao Liang for his encouragement and valuable suggestions. The authors would also like to thank the referees for a thoughtful reading.
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Han, X.L., Shao, M.Q. p-Laplacian Equations on Locally Finite Graphs. Acta. Math. Sin.-English Ser. 37, 1645–1678 (2021). https://doi.org/10.1007/s10114-021-9523-5
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DOI: https://doi.org/10.1007/s10114-021-9523-5