Taiwanese Journal of Mathematics

Existence and Multiplicity of Solutions for a Class of $(p,q)$-Laplacian Equations in $\mathbb{R}^N$ with Sign-changing Potential

Nian Zhang and Gao Jia

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Abstract

In this paper, we use variational approaches to establish the existence of weak solutions for a class of $(p,q)$-Laplacian equations on $\mathbb{R}^N$, for $1 \lt q \lt p \lt q^{*} := Nq/(N-q)$, $p \lt N$, with a sign-changing potential function and a Carathéodory reaction term which do not satisfy the Ambrosetti-Rabinowitz type growth condition. By linking theorem with Cerami condition, the fountain theorem and dual fountain theorem with Cerami condition, we obtain some existence of weak solutions for the above equations under our considerations which are different from those used in related papers.

Article information

Source
Taiwanese J. Math., Advance publication (2019), 20 pages.

Dates
First available in Project Euclid: 1 April 2019

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1554105651

Digital Object Identifier
doi:10.11650/tjm/190302

Subjects
Primary: 35A15: Variational methods 35J62: Quasilinear elliptic equations

Keywords
$(p,q)$-Laplacian Cerami condition linking theorem fountain theorem dual fountain theorem

Citation

Zhang, Nian; Jia, Gao. Existence and Multiplicity of Solutions for a Class of $(p,q)$-Laplacian Equations in $\mathbb{R}^N$ with Sign-changing Potential. Taiwanese J. Math., advance publication, 1 April 2019. doi:10.11650/tjm/190302. https://projecteuclid.org/euclid.twjm/1554105651


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References

  • A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Functional Analysis 14 (1973), no. 4, 349–381.
  • R. Bartolo, Multiplicity results for a class of quasilinear elliptic problems, Mediterr. J. Math. 11 (2014), no. 4, 1099–1113.
  • R. Bartolo, A. M. Candela and A. Salvatore, On a class of superlinear $(p,q)$-Laplacian type equations on $\mathbb{R}^{N}$, J. Math. Anal. Appl. 438 (2016), no. 1, 29–41.
  • ––––, Multiplicity results for a class of asymptotically $p$-linear equations on $\mathbb{R}^{N}$, Commun. Contemp. Math. 18 (2016), no. 1, 1550031, 24 pp.
  • T. Bartsch and Z. Q. Wang, Existence and multiplicity results for some superlinear elliptic problems on $\mathbb{R}^{N}$, Comm. Partial Differential Equations 20 (1995), no. 9-10, 1725–1741.
  • V. Benci and D. Fortunato, Discreteness conditions of the spectrum of Schrödinger operators, J. Math. Anal. Appl. 64 (1978), no. 3, 695–700.
  • H. Berestycki and P.-L. Lions, Nonlinear scalar field equations I: Existence of a ground state; II: Existence of infinitely many solutions, Arch. Rational Mech. Anal. 82 (1983), no. 4, 313–375.
  • H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, New York, 2011.
  • M. F. Chaves, G. Ercole and O. H. Miyagaki, Existence of a nontrivial solution for the $(p,q)$-Laplacian in $\mathbb{R}^{N}$ without the Ambrosetti-Rabinowitz condition, Nonlinear Anal. 114 (2015), 133–141.
  • D. G. Costa and O. H. Miyagaki, Nontrivial solutions for perturbations of the $p$-Laplacian on unbounded domains, J. Math. Anal. Appl. 193 (1995), no. 3, 737–755.
  • W. Y. Ding and W.-M. Ni, On the existence of positive entire solutions of a semilinear elliptic equation, Arch. Rational Mech. Anal. 91 (1986), no. 4, 283–308.
  • M. Fabian, P. Habala, P. Hájek, V. Montesinos and V. Zizler, Banach Space Theory: The basis for linear and nonlinear analysis, Springer, New York, 2011.
  • E. J. Hurtado, O. H. Miyagaki and R. S. Rodrigues, Existence and multiplicity of solutions for a class of elliptic equations without Ambrosetti-Rabinowitz type conditions, J. Dynam. Differential Equations 30 (2018), no. 2, 405–432.
  • G. Li and X. Liang, The existence of nontrivial solutions to nonlinear elliptic equation of $p$-$q$-Laplacian type on $\mathbb{R}^{N}$, Nonlinear Anal. 71 (2009), no. 5-6, 2316–2334.
  • G. Li and H.-S. Zhou, The existence of a positive solution to asymptotically linear scalar field equations, Proc. Roy. Soc. Edinburgh Sect. A 130 (2000), no. 1, 81–105.
  • S. Liu, On superlinear problems without the Ambrosetti and Rabinowitz condition, Nonlinear Anal. 73 (2010), no. 3, 788–795.
  • O. H. Miyagaki and M. A. S. Souto, Superlinear problems without Ambrosetti and Rabinowitz growth condition, J. Differential Equations 245 (2008), no. 12, 3628–3638.
  • D. Mugnai and N. S. Papageorgiou, Wang's multiplicity result for superlinear $(p,q)$-equations without the Ambrosetti-Rabinowitz condition, Trans. Amer. Math. Soc. 366 (2014), no. 9, 4919–4937.
  • N. S. Papageorgiou and V. D. Rădulescu, Resonant $(p,2)$-equations with asymmetric reaction, Anal. Appl. (Singap.) 13 (2015), no. 5, 481–506.
  • P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Regional Conference Series in Mathematics 65, American Mathematical Society, Providence, RI, 1986.
  • ––––, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys. 43 (1992), no. 2, 270–291.
  • A. Salvatore, Multiple solutions for perturbed elliptic equations in unbounded domains, Adv. Nonlinear Stud. 3 (2003), no. 1, 1–23.
  • L. Shao and H. Chen, Existence and concentration result for a quasilinear Schrödinger equation with critical growth, Z. Angew. Math. Phys. 68 (2017), no. 6, Art. 126, 16 pp.
  • M. Struwe, Variational Methods: Applications to nonlinear partial differential equations and Hamiltonian systems, Fourth edition, A Series of Modern Surveys in Mathematics 34, Springer-Verlag, Berlin, 2008.
  • K. Teng and C. Zhang, Infinitely many solutions for quasilinear elliptic equations involving $(p,q)$-Laplacian in $\mathbb{R}^{N}$, Nonlinear Anal. Real World Appl. 32 (2016), 242–259.
  • M. Willem, Minimax Theorems, Progress in Nonlinear Differential Equations and their Applications 24, Birkhäuser Boston, Boston, MA, 1996.
  • M. Wu and Z. Yang, A class of $p$-$q$-Laplacian type equation with potentials eigenvalue problem in $\mathbb{R}^{N}$, Bound. Value Probl. 2009 (2009), Art. ID 185319, 19 pp.
  • J. F. Yang and X. P. Zhu, On the existence of nontrivial solution of a quasilinear elliptic boundary value problem for unbounded domians I: Positive mass case, Acta. Math. Sci. (English Ed.) 7 (1987), no. 3, 341–359.
  • J. F. Zhao, Structure Theory of Banach Spaces, Wuhan University Press, Wuhan, 1991.