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Infinitely many solutions of strongly degenerate Schrödinger elliptic equations with vanishing potentials

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Abstract

In this paper, we are concerned with the existence of infinitely many nontrivial solutions to the following semilinear degenerate elliptic equation

$$\begin{aligned} -\Delta _\lambda u + V(x) u = f(x,u) \quad \text { in } {\mathbb {R}}^N, N\ge 3, \end{aligned}$$

where \(V: {\mathbb {R}}^N\rightarrow {\mathbb {R}}\) is a potential function and allowed to be vanishing at infinitely, \(f: {\mathbb {R}}^N\times {\mathbb {R}}\rightarrow {\mathbb {R}}\) is a given function and \(\Delta _\lambda \) is the strongly degenerate elliptic operator. Under suitable assumptions on the potential V and the nonlinearity f,  some results on the multiplicity of solutions are proved. The proofs are based on variational methods, in particular, on the well-known mountain pass lemma of Ambrosetti–Rabinowitz. Due to the vanishing potentials and degeneracy of the operator, some new compact embedding theorems are used in the proof. Our results extend and generalize some existing results (Alves and Souto in J Differ Equ 254:1977–1991, 2013; Hamdani in Asia-Eur J Math 13:2050131, https://doi.org/10.1142/S1793557120501314, 2020; Luyen in Commun Math Anal 22:61–75, 2019; Luyen and Tri in J Math Anal Appl 461:1271–1286, 2018; Tang in J Math Anal Appl 401:407–415, 2013; Toon and Ubilla in Discrete Contin Dyn Syst 40:5831–5843, 2020).

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References

  1. Anh, C.T.: Global attractor for a semilinear strongly degenerate parabolic equation on \({\mathbb{R} }^N\), NoDEA. Nonlinear Differ. Equ. Appl. 21, 663–678 (2014)

    Article  Google Scholar 

  2. Alves, C.O., Souto, M.A.S.: Existence of solutions for a class of nonlinear Schrödinger equations with potential vanishing at infinity. J. Differ. Equ. 254, 1977–1991 (2013)

    Article  Google Scholar 

  3. Ambrosetti, A., Felli, V., Malchiodi, A.: Ground states of nonlinear Schrödinger equations with potentials vanishing at infinity. J. Eur. Math. Soc. (JEMS) 7, 117–144 (2005)

    Article  MathSciNet  Google Scholar 

  4. Ambrosetti, A., Wang, Z.Q.: Nonlinear Schrödinger equations with vanishing and decaying potentials. Differ. Integral Equ. 18, 1321–1332 (2005)

    Google Scholar 

  5. Anh, C.T., My, B.K.: Existence of solutions to \(\Delta _\lambda \)-Laplace equations without the Ambrosetti–Rabinowitz condition. Complex Var. Elliptic Equ. 61, 137–150 (2016)

    Article  MathSciNet  Google Scholar 

  6. Anh, C.T., My, B.K.: Liouville type theorems for elliptic inequalities involving the \(\Delta _\lambda \)-Laplace operator. Complex Var. Elliptic Equ. 61, 1002–1013 (2016)

    Article  MathSciNet  Google Scholar 

  7. Anh, C.T., My, B.K.: Existence and non-existence of solutions to a Hamiltonian strongly degenerate elliptic system. Adv. Nonlinear Anal. 8, 661–678 (2019)

    Article  MathSciNet  Google Scholar 

  8. Anh, C.T., Lee, J., My, B.K.: On a class of Hamiltonian strongly degenerate elliptic systems with concave and convex nonlinearities. Complex Var. Elliptic Equ. 65, 648–671 (2020)

    Article  MathSciNet  Google Scholar 

  9. Benci, V., Fortunato, D.: Variational Methods in Nonlinear Field Equations. Solitary Waves, Hylomorphic Solitons and Vortices. Springer Monographs in Mathematics, Springer, Cham (2014)

    Book  Google Scholar 

  10. Chen, J., Tang, X., Gao, Z.: Infinitely many solutions for semilinear \(\Delta _\lambda \)-Laplace equations with sign-changing potential and nonlinearity. Studia Sci. Math. Hungarica. 54, 536–549 (2017)

    MathSciNet  Google Scholar 

  11. Ding, Y., Lee, C.: Multiple solutions of Schrödinger equations with indefinite linear part and super or asymptotically linear terms. J. Differ. Equ. 222, 137–163 (2006)

    Article  Google Scholar 

  12. Franchi, B., Lanconelli, E.: Une métrique associée à une classe d’copérateurs elliptiques dégénérés, (French) [A metric associated with a class of degenerate elliptic operators] Conference on linear partial and pseudodifferential operators (Torino, 1982). Rend Sem Mat Univ Politec Torino1983. Special Issue, pp. 105–114 (1984)

  13. Han, Q.: Compact embedding results of Sobolev spaces and existence of positive solutions to quasilinear equations. Bull. Sci. Math. 141, 46–71 (2017)

    Article  MathSciNet  Google Scholar 

  14. Hamdani, M.K.: Multiple solutions for Grushin operator without odd nonlinearity. Asia-Eur. J. Math. 13, 2050131 (2020). https://doi.org/10.1142/S1793557120501314

    Article  MathSciNet  Google Scholar 

  15. Kogoj, A.E., Lancenolli, E.: On semilinear \(\Delta _\lambda \)-Laplace equation. Nonlinear Anal. 75, 4637–4649 (2012)

    Article  MathSciNet  Google Scholar 

  16. Kogoj, A.E., Lanconelli, E.: Linear and semilinear problems involving \(\Delta _\lambda \)-Laplacians. Electron. J. Differ. Equ. 25, 167–178 (2018)

    MathSciNet  Google Scholar 

  17. Luyen, D.T.: Multiple solutions for semilinear \(\Delta _\gamma \) differential equations in \({\mathbb{R} }^N\) with sign-changing potential. Commun. Math. Anal. 22, 61–75 (2019)

    MathSciNet  Google Scholar 

  18. Luyen, D.T., Tri, N.M.: Existence of infinitely many solutions for semilinear degenerate Schrödinger equations. J. Math. Anal. Appl. 461, 1271–1286 (2018)

    Article  MathSciNet  Google Scholar 

  19. Rabinowitz, P.H.: Minimax methods in critical point theory with applications to differential equations. In: CBMS Regional Conference Series in Mathematics, vol. 65, American Mathematics Society, Providence (1986)

  20. Rahal, B., Hamdani, M.K.: Infinitely many solutions for \(\Delta _\alpha \)-Laplace equations with sign-changing potential. J. Fixed Point Theory Appl. 20, 137 (2018). https://doi.org/10.1007/s11784-018-0617-3

    Article  MathSciNet  Google Scholar 

  21. Tang, X.: Infinitely many solutions for semilinear Schrödinger equations with sign-changing potential and nonlinearity. J. Math. Anal. Appl. 401, 407–415 (2013)

    Article  MathSciNet  Google Scholar 

  22. Thuy, P.T., Tri, N.M.: Nontrivial solutions to boundary value problems for semilinear strongly degenerate elliptic differential equations, NoDAE. Nonlinear Differ. Equ. Appl. 19, 279–298 (2012)

    Article  Google Scholar 

  23. Toon, E., Ubilla, P.: Existence of positive solutions of Schrödinger equations with vanishing potentials. Discrete Contin. Dyn. Syst. 40, 5831–5843 (2020)

    Article  MathSciNet  Google Scholar 

  24. Toon, E., Ubilla, P.: Hamiltonian systems of Schrödinger equations with vanishing potentials. Commun. Contemp. Math. 24(1), Paper No. 2050074 (2022). https://doi.org/10.1142/S0219199720500741

  25. Yang, Y.: Solitons in Field Theory and Nonlinear Analysis, Springer Monographs in Mathematics. Springer, New York (2001)

    Google Scholar 

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Acknowledgements

The author would like to thank the referees for their helpful comments and suggestions, which improved the presentation of this paper.

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  1. Faculty of Primary Education, Hanoi Pedagogical University 2, 32 Nguyen Van Linh, Xuan Hoa, Phuc Yen, Vinh Phuc, Vietnam

    Bui Kim My

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My, B.K. Infinitely many solutions of strongly degenerate Schrödinger elliptic equations with vanishing potentials. Anal.Math.Phys. 14, 43 (2024). https://doi.org/10.1007/s13324-024-00903-4

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  • DOI: https://doi.org/10.1007/s13324-024-00903-4

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