Opuscula Math. 40, no. 1 (2020), 93-110
https://doi.org/10.7494/OpMath.2020.40.1.93

 
Opuscula Mathematica

Fractional p&q-Laplacian problems with potentials vanishing at infinity

Teresa Isernia

Abstract. In this paper we prove the existence of a positive and a negative ground state weak solution for the following class of fractional \(p\&q\)-Laplacian problems \[\begin{aligned} (-\Delta)_{p}^{s} u + (-\Delta)_{q}^{s} u + V(x) (|u|^{p-2}u + |u|^{q-2}u)= K(x) f(u) \quad \text{ in } \mathbb{R}^{N},\end{aligned}\] where \(s\in (0, 1)\), \(1\lt p\lt q \lt\frac{N}{s}\), \(V: \mathbb{R}^{N}\to \mathbb{R}\) and \(K: \mathbb{R}^{N}\to \mathbb{R}\) are continuous, positive functions, allowed for vanishing behavior at infinity, \(f\) is a continuous function with quasicritical growth and the leading operator \((-\Delta)^{s}_{t}\), with \(t\in \{p,q\}\), is the fractional \(t\)-Laplacian operator.

Keywords: fractional \(p\&q\)-Laplacian, vanishing potentials, ground state solution.

Mathematics Subject Classification: 35A15, 35J60, 35R11, 45G05.

Full text (pdf)

  1. C.O. Alves, M.A.S. Souto, Existence of solutions for a class of elliptic equations in \(\mathbb{R}^N\) with vanishing potentials, J. Differential Equations 252 (2012), 5555-5568.
  2. C.O. Alves, M.A.S. Souto, Existence of solutions for a class of nonlinear Schrödinger equations with potential vanishing at infinity, J. Differential Equations 254 (2013), 1977-1991.
  3. C.O. Alves, V. Ambrosio, T. Isernia, Existence, multiplicity and concentration for a class of fractional \(p\&q\) Laplacian problems in \(\mathbb{R}^N\), Commun. Pure Appl. Anal. 18 (2019) 4, 2009-2045.
  4. A. Ambrosetti, V. Felli, A. Malchiodi, Ground states of nonlinear Schrödinger equations with potentials vanishing at infinity, J. Eur. Math. Soc. (JEMS) 7 (2005) 1, 117-144.
  5. A. Ambrosetti, P.H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal. 14 (1973), 349-381.
  6. V. Ambrosio, A multiplicity result for a fractional \(p\)-laplacian problem without growth conditions, Riv. Mat. Univ. Parma 9 (1) (2018), 53-71.
  7. V. Ambrosio, Concentrating solutions for a class of nonlinear fractional Schrödinger equations in \(\mathbb{R}^N\), Rev. Mat. Iberoam. 35 (2019) 5, 1367-1414.
  8. V. Ambrosio, Fractional \(p\&q\) Laplacian problems in \(\mathbb{R}^N\) with critical growth, Z. Anal. Anwend. 32 (2019), 301-327.
  9. V. Ambrosio, T. Isernia, Sign-changing solutions for a class of Schrödinger equations with vanishing potentials, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 29 (2018) 1, 127-152.
  10. V. Ambrosio, T. Isernia, Multiplicity and concentration results for some nonlinear Schrödinger equations with the fractional \(p\)-Laplacian, Discrete Contin. Dyn. Syst. 38 (11) (2018), 5835-5881.
  11. V. Ambrosio, T. Isernia, On a fractional \(p\&q\) Laplacian problem with critical Sobolev-Hardy exponents, Mediterr. J. Math. 15 (2018) 6, Art. 219, 17 pp.
  12. V. Ambrosio, T. Isernia, On the multiplicity and concentration for \(p\)-fractional Schrödinger equations, Appl. Math. Lett. 95 (2019), 13-22.
  13. V. Ambrosio, G.M. Figueiredo, T. Isernia, G. Molica Bisci, Sign-changing solutions for a class of zero mass nonlocal Schrödinger equations, Adv. Nonlinear Stud. 19 (2019) 1, 113-132.
  14. V. Ambrosio, T. Isernia, G. Siciliano, On a fractional \(p\&q\) Laplacian problem with critical growth, Minimax Theory Appl. 4 (2019) 1, 1-19.
  15. S. Barile, G.M. Figueiredo, Existence of a least energy nodal solution for a class of \(p\&q\)-quasilinear elliptic equations, Adv. Nonlinear Stud. 14 (2014) 2, 511-530.
  16. S. Barile, G.M. Figueiredo, Existence of least energy positive, negative and nodal solutions for a class of \(p\&q\)-problems with potentials vanishing at infinity, J. Math. Anal. Appl. 427 (2015) 2, 1205-1233.
  17. V. Benci, C.R. Grisanti, A.M. Micheletti, Existence of solutions for the nonlinear Schrödinger equation with \(V(\infty)=0\), Progr. Nonlinear Differential Equations Appl. 66 (2005), 53-65.
  18. H. Berestycki, P.L. Lions, Nonlinear scalar field equations. I. Existence of a ground state, Arch. Ration. Mech. Anal. 82 (1983), 313-346.
  19. M. Bhakta, D. Mukherjee, Multiplicity results for \((p,q)\) fractional elliptic equations involving critical nonlinearities, Adv. Differential Equations 24 (2019) 3-4, 185-228.
  20. D. Bonheure, J. Van Schaftingen, Ground states for the nonlinear Schrödinger equation with potential vanishing at infinity, Ann. Mat. Pura Appl. (4) 189 (2010) 2, 273-301.
  21. D. Borkowski, K. Jańczak-Borkowska, Backward stochastic variational inequalities driven by multidimensional fractional Brownian motion, Opuscula Math. 38 (2018) 3, 307-326.
  22. H. Brezis, E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc. 88 (1983) 3, 486-490.
  23. L. Caffarelli, Non-local diffusions, drifts and games, [in:] Nonlinear Partial Differential Equations, volume 7 of Abel Symposia, pp. 37-52, 2012.
  24. L. Caffarelli, L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations 32 (2007), 1245-1260.
  25. C. Chen, J. Bao, Existence, nonexistence, and multiplicity of solutions for the fractional \(p\&q\)-Laplacian equation in \(\mathbb{R}^N\), Bound. Value Probl. 2016, Paper No. 153, 16 pp.
  26. L. Cherfils, V. Il'yasov, On the stationary solutions of generalized reaction difusion equations with \(p\&q\)-Laplacian, Commun. Pure Appl. Anal. 1 (2004), 1-14.
  27. A. Di Castro, T. Kuusi, G. Palatucci, Local behavior of fractional \(p\)-minimizers, Ann. Inst. H. Poincarè Anal. Non Linèaire 33 (5) (2016), 1279-1299.
  28. E. Di Nezza, G. Palatucci, E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. math. 136 (2012), 521-573.
  29. P. Felmer, A. Quaas, J. Tan, Positive solutions of the nonlinear Schrödinger equation with the fractional Laplacian, Proc. Roy. Soc. Edinburgh Sect. A 142 (2012), 1237-1262.
  30. G.M. Figueiredo, Existence of positive solutions for a class of \(p\&q\) elliptic problems with critical growth on \(\mathbb{R}^N\), J. Math. Anal. Appl. 378 (2011), 507-518.
  31. G. Franzina, G. Palatucci, Fractional \(p\)-eigenvalues, Riv. Math. Univ. Parma (N.S.) 5 (2014), 373-386.
  32. C. He, G. Li, The regularity of weak solutions to nonlinear scalar field elliptic equations containing \(p\&q\)-Laplacians, Ann. Acad. Sci. Fenn. Math. 33 (2008) 2, 337-371.
  33. T. Isernia, On a nonhomogeneous sublinear-superlinear fractional equation in \(\mathbb{R}^N\), Riv. Math. Univ. Parma (N.S.) 10 (2019) 1, 167-186.
  34. T. Isernia, Sign-changing solutions for a fractional Kirchhoff equation, Nonlinear Anal. 190 (2020), 111623, 20 pp.
  35. N. Laskin, Fractional quantum mechanics and Lévy path integrals, Phys. Lett. A 268 (2000) 4-6, 298-305.
  36. G. Li, Z. Guo, Multiple solutions for the \(p\&q\)-Laplacian problem with critical exponent, Acta Math. Sci. Ser. B Engl. Ed. 29 (4) (2009), 903-918.
  37. G.B. Li, X. Liang, The existence of nontrivial solutions to nonlinear elliptic equation of \(p-q\)-Laplacian type on \(\mathbb{R}^N\), Nonlinear Anal. 71 (2009), 2316-2334.
  38. E. Lindgren, P. Lindqvist, Fractional eigenvalues, Calc. Var. 49 (2014), 795-826.
  39. G. Molica Bisci, V. Rădulescu, Ground state solutions of scalar field fractional Schrödinger equations, Calc. Var. Partial Differential Equations 54 (2015) 3, 2985-3008.
  40. G. Molica Bisci, V. Rădulescu, R. Servadei, Variational Methods for Nonlocal Fractional Problems, Cambridge University Press, 162, Cambridge, 2016.
  41. T. Mukherjee, K. Sreenadh, On Dirichlet problem for fractional \(p\)-Laplacian with singular non-linearity, Adv. Nonlinear Anal. 8 (2019) 1, 52-72.
  42. N. Papageorgiou, V. Rădulescu, D. Repovš, Ground state and nodal solutions for a class of double phase problems, Z. Angew. Math. Phys. 71 (2020) 1, Paper No. 15.
  43. S. Secchi, Ground state solutions for nonlinear fractional Schrödinger equations in \(\mathbb{R}^N\), J. Math. Phys. 54 (2013), 031501.
  44. M. Willem, Minimax Theorems, Progress in Nonlinear Differential Equations and their Applications, 24. Birkhäuser Boston, Inc., Boston, MA, 1996.
  • Teresa Isernia
  • ORCID iD https://orcid.org/0000-0002-6215-3219
  • Dipartimento di Ingegneria Industriale e Scienze Matematiche, Università Politecnica delle Marche, Via Brecce Bianche, 12, 60131 Ancona, Italy
  • Communicated by Patrizia Pucci.
  • Received: 2019-12-15.
  • Revised: 2020-01-19.
  • Accepted: 2020-01-21.
  • Published online: 2020-02-17.
Opuscula Mathematica - cover

Cite this article as:
Teresa Isernia, Fractional p&q-Laplacian problems with potentials vanishing at infinity, Opuscula Math. 40, no. 1 (2020), 93-110, https://doi.org/10.7494/OpMath.2020.40.1.93

Download this article's citation as:
a .bib file (BibTeX),
a .ris file (RefMan),
a .enw file (EndNote)
or export to RefWorks.

In accordance with EU legislation we advise you this website uses cookies to allow us to see how the site is used. All data is anonymized.
All recent versions of popular browsers give users a level of control over cookies. Users can set their browsers to accept or reject all, or certain, cookies.