Communications of the Korean Mathematical Society (대한수학회논문집)

Korean Mathematical Society (대한수학회)
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EXISTENCE AND NONEXISTENCE OF SOLUTIONS FOR A CLASS OF HAMILTONIAN STRONGLY DEGENERATE ELLIPTIC SYSTEM

  • Received : 2022.03.24
  • Accepted : 2023.03.16
  • Published : 2023.07.31
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Abstract

In this paper, we study the existence and nonexistence of solutions for a class of Hamiltonian strongly degenerate elliptic system with subcritical growth $$\left{\array{-{\Delta}_{\lambda}u-{\mu}v={\mid}v{\mid}^{p-1}v&&\text{in }{\Omega},\\-{\Delta}_{\lambda}v-{\mu}u={\mid}u{\mid}^{q-1}u&&\text{in }{\Omega},\\u=v=0&&\text{ on }{\partial}{\Omega},}$$ where p, q > 1 and Ω is a smooth bounded domain in ℝN, N ≥ 3. Here Δλ is the strongly degenerate elliptic operator. The existence of at least a nontrivial solution is obtained by variational methods while the nonexistence of positive solutions are proven by a contradiction argument.

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References

  1. C. T. Anh, J. Lee, and B. K. My, On a class of Hamiltonian strongly degenerate elliptic systems with concave and convex nonlinearities, Complex Var. Elliptic Equ. 65 (2020), no. 4, 648-671. https://doi.org/10.1080/17476933.2019.1608971
  2. C. T. Anh and B. K. My, Existence of solutions to Δλ-Laplace equations without the Ambrosetti-Rabinowitz condition, Complex Var. Elliptic Equ. 61 (2016), no. 1, 137-150. https://doi.org/10.1080/17476933.2015.1068762
  3. C. T. Anh and B. K. My, Liouville-type theorems for elliptic inequalities involving the Δλ-Laplace operator, Complex Var. Elliptic Equ. 61 (2016), no. 7, 1002-1013. https://doi.org/10.1080/17476933.2015.1131685
  4. C. T. Anh and B. K. My, Existence and non-existence of solutions to a Hamiltonian strongly degenerate elliptic system, Adv. Nonlinear Anal. 8 (2019), no. 1, 661-678. https://doi.org/10.1515/anona-2016-0165
  5. S. Barile and A. Salvatore, Existence and multiplicity results for some Lane-Emden elliptic systems: subquadratic case, Adv. Nonlinear Anal. 4 (2015), no. 1, 25-35. https://doi.org/10.1515/anona-2014-0049
  6. T. Bartsch and D. G. de Figueiredo, Infinitely many solutions of nonlinear elliptic systems, in Topics in nonlinear analysis, 51-67, Progr. Nonlinear Differential Equations Appl., 35, Birkhauser, Basel, 1999.
  7. D. Bonheure, E. Moreira dos Santos, and M. Ramos, Ground state and non-ground state solutions of some strongly coupled elliptic systems, Trans. Amer. Math. Soc. 364 (2012), no. 1, 447-491. https://doi.org/10.1090/S0002-9947-2011-05452-8
  8. D. Bonheure, E. Moreira dos Santos, and H. Tavares, Hamiltonian elliptic systems: a guide to variational frameworks, Port. Math. 71 (2014), no. 3-4, 301-395. https://doi.org/10.4171/PM/1954
  9. J. Chen, X. Tang, and Z. Gao, Infinitely many solutions for semilinear Δλ-Laplace equations with sign-changing potential and nonlinearity, Studia Sci. Math. Hungar. 54 (2017), no. 4, 536-549. https://doi.org/10.1556/012.2017.54.4.1382
  10. P. Cl'ement, D. G. de Figueiredo, and E. L. Mitidieri, Positive solutions of semilinear elliptic systems, Comm. Partial Differential Equations 17 (1992), no. 5-6, 923-940. https://doi.org/10.1080/03605309208820869
  11. P. L. Felmer and S. Mart'inez, Existence and uniqueness of positive solutions to certain differential systems, Adv. Differential Equations 3 (1998), no. 4, 575-593.
  12. D. G. de Figueiredo and P. L. Felmer, On superquadratic elliptic systems, Trans. Amer. Math. Soc. 343 (1994), no. 1, 99-116. https://doi.org/10.2307/2154523
  13. D. G. de Figueiredo and B. Ruf, Elliptic systems with nonlinearities of arbitrary growth, Mediterr. J. Math. 1 (2004), no. 4, 417-431. https://doi.org/10.1007/s00009-004-0021-7
  14. B. Franchi and E. Lanconelli, Une metrique associee a une classe d'operateurs elliptiques degeneres, Rend. Sem. Mat. Univ. Politec. Torino 1983, Special Issue, 105-114 (1984).
  15. J. Hulshof and R. C. A. M. Van der Vorst, Differential systems with strongly indefinite variational structure, J. Funct. Anal. 114 (1993), no. 1, 32-58. https://doi.org/10.1006/jfan.1993.1062
  16. A. E. Kogoj and E. Lanconelli, On semilinear Δλ-Laplace equation, Nonlinear Anal. 75 (2012), no. 12, 4637-4649. https://doi.org/10.1016/j.na.2011.10.007
  17. A. E. Kogoj and E. Lanconelli, Linear and semilinear problems involving Δλ-Laplacians, in Proceedings of the International Conference "Two nonlinear days in Urbino 2017", 167-178, Electron. J. Differ. Equ. Conf., 25, Texas State Univ.-San Marcos, Dept. Math., San Marcos, TX, 2018.
  18. S. Li and M. Willem, Applications of local linking to critical point theory, J. Math. Anal. Appl. 189 (1995), no. 1, 6-32. https://doi.org/10.1006/jmaa.1995.1002
  19. D. T. Luyen and N. M. Tri, Existence of infinitely many solutions for semilinear degenerate Schrodinger equations, J. Math. Anal. Appl. 461 (2018), no. 2, 1271-1286. https://doi.org/10.1016/j.jmaa.2018.01.016
  20. E. L. Mitidieri, A Rellich type identity and applications, Comm. Partial Differential Equations 18 (1993), no. 1-2, 125-151. https://doi.org/10.1080/03605309308820923
  21. B. K. My, Nontrivial solutions for a class of Hamiltonian strongly degenerate elliptic system, Appl. Anal. (2022). https://doi.org/10.1080/00036811.2022.2027376
  22. B. K. My, On the existence of solutions of Hamiltonian strongly degenerate elliptic system with potentials in RN , Z. Anal. Anwend. (2023). https://doi.org/10.4171/ZAA/1717
  23. B. Rahal and M. K. Hamdani, Infinitely many solutions for Δα-Laplace equations with sign-changing potential, J. Fixed Point Theory Appl. 20 (2018), no. 4, Paper No. 137, 17 pp. https://doi.org/10.1007/s11784-018-0617-3
  24. B. Ruf, Superlinear elliptic equations and systems, in Handbook of differential equations: stationary partial differential equations. Vol. V, 211-276, Handb. Differ. Equ, Elsevier/North-Holland, Amsterdam, 2008. https://doi.org/10.1016/S1874-5733(08)80010-1
  25. P. T. Thuy and N. M. Tri, Nontrivial solutions to boundary value problems for semilinear strongly degenerate elliptic differential equations, NoDEA Nonlinear Differential Equations Appl. 19 (2012), no. 3, 279-298. https://doi.org/10.1007/s00030-011-0128-z
  26. R. C. A. M. Van der Vorst, Variational identities and applications to differential systems, Arch. Rational Mech. Anal. 116 (1992), no. 4, 375-398. https://doi.org/10.1007/BF00375674