DOI QR코드

DOI QR Code

ON A CLASS OF QUASILINEAR ELLIPTIC EQUATION WITH INDEFINITE WEIGHTS ON GRAPHS

  • Man, Shoudong ;
  • Zhang, Guoqing
  • Received : 2018.07.04
  • Accepted : 2019.04.01
  • Published : 2019.07.01

Abstract

Suppose that G = (V, E) is a connected locally finite graph with the vertex set V and the edge set E. Let ${\Omega}{\subset}V$ be a bounded domain. Consider the following quasilinear elliptic equation on graph G $$\{-{\Delta}_{pu}={\lambda}K(x){\mid}u{\mid}^{p-2}u+f(x,u),\;x{\in}{\Omega}^{\circ},\\u=0,\;x{\in}{\partial}{\Omega},$$ where ${\Omega}^{\circ}$ and ${\partial}{\Omega}$ denote the interior and the boundary of ${\Omega}$, respectively, ${\Delta}_p$ is the discrete p-Laplacian, K(x) is a given function which may change sign, ${\lambda}$ is the eigenvalue parameter and f(x, u) has exponential growth. We prove the existence and monotonicity of the principal eigenvalue of the corresponding eigenvalue problem. Furthermore, we also obtain the existence of a positive solution by using variational methods.

Keywords

indefinite weights;quasilinear elliptic equation on graphs;eigenvalue problem on graphs

References

  1. A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Functional Analysis 14 (1973), 349-381.
  2. M. Degiovanni and S. Lancelotti, Linking over cones and nontrivial solutions for p-Laplace equations with p-superlinear nonlinearity, Ann. Inst. H. Poincare Anal. Non Lineaire 24 (2007), no. 6, 907-919.
  3. X. Fan and Z. Li, Linking and existence results for perturbations of the p-Laplacian, Nonlinear Anal. 42 (2000), no. 8, 1413-1420. https://doi.org/10.1016/S0362-546X(99)00161-3
  4. H. Ge, A p-th Yamabe equation on graph, Proc. Amer. Math. Soc. 146 (2018), no. 5, 2219-2224.
  5. A. Grigor'yan, Y. Lin, and Y. Yang, Kazdan-Warner equation on graph, Calc. Var. Partial Differential Equations 55 (2016), no. 4, Art. 92, 13 pp.
  6. A. Grigor'yan, Y. Lin, and Y. Yang, Yamabe type equations on graphs, J. Differential Equations 261 (2016), no. 9, 4924-4943. https://doi.org/10.1016/j.jde.2016.07.011
  7. A. Grigor'yan, Y. Lin, and Y. Yang, Existence of positive solutions to some nonlinear equations on locally finite graphs, Sci. China Math. 60 (2017), no. 7, 1311-1324. https://doi.org/10.1007/s11425-016-0422-y
  8. Y. Lin and Y. Wu, The existence and nonexistence of global solutions for a semilinear heat equation on graphs, Calc. Var. Partial Differential Equations 56 (2017), no. 4, Art. 102, 22 pp. https://doi.org/10.1007/s00526-017-1114-z
  9. J. Moser, A sharp form of an inequality by N. Trudinger, Indiana Univ. Math. J. 20 (1970/71), 1077-1092.
  10. B. Xuan, Existence results for a superlinear p-Laplacian equation with indefinite weights, Nonlinear Anal. 54 (2003), no. 5, 949-958. https://doi.org/10.1016/S0362-546X(03)00120-2
  11. Y. Yang, Trudinger-Moser inequalities on complete noncompact Riemannian manifolds, J. Funct. Anal. 263 (2012), no. 7, 1894-1938. https://doi.org/10.1016/j.jfa.2012.06.019
  12. D. Zhang, Semi-linear elliptic equations on graphs, J. Partial Differ. Equ. 30 (2017), no. 3, 221-231.
  13. G. Q. Zhang and S. Y. Liu, An elliptic equation with critical potential and indefinite weights in R2, Acta Math. Sci. Ser. A (Chin. Ed.) 28 (2008), no. 5, 929-936.
  14. N. Zhang and L. Zhao, Convergence of ground state solutions for nonlinear Schrodinger equations on graphs, Sci. China Math. 61 (2018), no. 8, 1481-1494.
  15. X. Zhang and A. Lin, Positive solutions of p-th Yamabe type equations on infinite graphs, Proc. Amer. Math. Soc. 147 (2019), no. 4, 1421-1427.
  16. H. Ge and W. Jiang, The 1-Yamabe equation on graph, Commun. in Contemporary Math. 20 (2018), no 5, 32-38.

Acknowledgement

Supported by : National Natural Science Foundation of China