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Concentrating solutions for a fractional $p$ -Laplacian logarithmic Schrödinger equation

Claudianor O. Alves

https://orcid.org/0000-0001-6921-7548

Universidade Federal de Campina Grande, Unidade Acadêmica de Matemática, CEP 58429-900 Campina Grande, Paraíba, Brazil

E-mail Address: coalves@mat.ufcg.edu.br

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and

Vincenzo Ambrosio

https://orcid.org/0000-0003-3439-1428

Dipartimento di Ingegneria Industriale e Scienze Matematiche, Università Politecnica delle Marche, Via Brecce Bianche, 12, 60131 Ancona, Italy

E-mail Address: v.ambrosio@staff.univpm.it

Corresponding author.

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https://doi.org/10.1142/S0219530523500288Cited by:0 (Source: Crossref)

Abstract

We consider the following fractional $p$ -Laplacian logarithmic Schrödinger equation:

where

$𝜖 > 0$ ,

$s \in (0, 1)$ ,

$p \in [2, \infty)$ ,

$N > s p$ ,

${(- Δ)}_{p}^{s}$ is the fractional

$p$ -Laplacian operator,

$V : ℝ^{N} \to ℝ$ is a continuous potential satisfying a local condition. By applying suitable variational arguments, we analyze the existence and concentration of solutions as

$𝜖 \to 0$ for the above problem.

Keywords:

AMSC: 35R11, 35A15, 35J10, 35B09

References

1. C. O. Alves and I. S. da Silva, Existence of multiple solutions for a Schrödinger logarithmic equation via Lusternik-Schnirelmann category, to appear in Analysis and Applications. Google Scholar
2. C. O. Alves and D. C. de Morais Filho, Existence and concentration of positive solutions for a Schrödinger logarithmic equation, Z. Angew. Math. Phys. 69(6) (2018) 144. Crossref, Web of Science, Google Scholar
3. C. O. Alves and C. Ji, Existence and concentration of positive solutions for a logarithmic Schrödinger equation via penalization method, Calc. Var. Partial Differential Equations 59(1) (2020) 21. Crossref, Web of Science, Google Scholar
4. C. O. Alves and C. Ji, Multi-peak positive solutions for a logarithmic Schrödinger equation via variational methods, Isr. J. Math. (2023), https://doi.org/10.1007/s11856-023-2494-8. Crossref, Web of Science, Google Scholar
5. C. O. Alves and O. H. Miyagaki, Existence and concentration of solution for a class of fractional elliptic equation in $ℝ^{N}$ via penalization method, Calc. Var. Partial Differential Equations 55(3) (2016) 47. Crossref, Web of Science, Google Scholar
6. A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal. 14 (1973) 349–381. Crossref, Google Scholar
7. V. Ambrosio, Concentration phenomena for a class of fractional Kirchhoff equations in $ℝ^{N}$ with general nonlinearities, Nonlinear Anal. 195 (2020) 111761. Crossref, Web of Science, Google Scholar
8. V. Ambrosio, Nonlinear Fractional Schrödinger Equations in ℝN (Birkhäuser, 2021). Crossref, Google Scholar
9. V. Ambrosio, On the uniform vanishing property at infinity of $W^{s, p}$ -sequences, Nonlinear Anal. 238 (2024) 113398. Crossref, Web of Science, Google Scholar
10. V. Ambrosio, G. M. Figueiredo and T. Isernia, Existence and concentration of positive solutions for $p$ -fractional Schrödinger equations, Ann. Mat. Pura Appl. (4) 199(1) (2020) 317–344. Crossref, Web of Science, Google Scholar
11. V. Ambrosio and 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. Crossref, Web of Science, Google Scholar
12. A. H. Ardila, Existence and stability of standing waves for nonlinear fractional Schrödinger equation with logarithmic nonlinearity, Nonlinear Anal. 155 (2017) 52–64. Crossref, Web of Science, Google Scholar
13. I. Bialynicki-Birula and J. Mycielski, Nonlinear wave mechanics, Ann. Physics 100(1–2) (1976) 62–93. Crossref, Web of Science, Google Scholar
14. C. Bucur and E. Valdinoci, Nonlocal Diffusion and Applications, Lecture Notes of the Unione Matematica Italiana, Vol. 20 (Springer, Cham, 2016), pp. xii+155. Crossref, Google Scholar
15. T. Cazenave, Stable solutions of the logarithmic Schrödinger equation, Nonlinear Anal. 7(10) (1983) 1127–1140. Crossref, Web of Science, Google Scholar
16. T. Cazenave and A. Haraux, Équations d’évolution avec non linéarité logarithmique, Ann. Fac. Sci. Toulouse Math. (5) 2(1) (1980) 21–51. Crossref, Google Scholar
17. A. Cotsiolis and N. K. Tavoularis, On logarithmic Sobolev inequalities for higher order fractional derivatives, C. R. Math. Acad. Sci. Paris 340(3) (2005) 205–208. Crossref, Web of Science, Google Scholar
18. P. d’Avenia, E. Montefusco and M. Squassina, On the logarithmic Schrödinger equation, Commun. Contemp. Math. 16(2) (2014) 1350032. Link, Web of Science, Google Scholar
19. P. d’Avenia, M. Squassina and M. Zenari, Fractional logarithmic Schrödinger equations, Math. Methods Appl. Sci. 38(18) (2015) 5207–5216. Crossref, Web of Science, Google Scholar
20. E. d. S. Böer and O. H. Miyagaki, Existence and multiplicity of solutions for the fractional $p$ -Laplacian Choquard logarithmic equation involving a nonlinearity with exponential critical and subcritical growth, J. Math. Phys. 62(5) (2021) 051507. Crossref, Web of Science, Google Scholar
21. L. M. Del Pezzo and A. Quaas, A Hopf’s lemma and a strong minimum principle for the fractional $p$ -Laplacian, J. Differential Equations 263(1) (2017) 765–778. Crossref, Web of Science, Google Scholar
22. L. M. Del Pezzo and A. Quaas, Spectrum of the fractional p-Laplacian in $ℝ^{N}$ and decay estimate for positive solutions of a Schrödinger equation, Nonlinear Anal. 193 (2020) 111479. Crossref, Web of Science, Google Scholar
23. M. Del Pino and J. Dolbeault, The optimal Euclidean $L^{p}$ -Sobolev logarithmic inequality, J. Funct. Anal. 197(1) (2003) 151–161. Crossref, Web of Science, Google Scholar
24. M. del Pino and P. L. Felmer, Local mountain passes for semilinear elliptic problems in unbounded domains, Calc. Var. Partial Differential Equations 4 (1996) 121–137. Crossref, Web of Science, Google Scholar
25. E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker’s guide to the fractional Sobolev spaces, Bull. Sci. Math. 136(5) (2012) 521–573. Crossref, Web of Science, Google Scholar
26. P. Felmer, A. Quaas and J. Tan, Positive solutions of the nonlinear Schrödinger equation with the fractional Laplacian, Proc. Roy. Soc. Edinburgh Sect. A 142(6) (2012) 1237–1262. Crossref, Web of Science, Google Scholar
27. G. M. Figueiredo and G. Siciliano, A multiplicity result via Ljusternick-Schnirelmann category and Morse theory for a fractional Schrödinger equation in $ℝ^{N}$ , NoDEA Nonlinear Differential Equations Appl. 23 (2016) 12. Crossref, Web of Science, Google Scholar
28. A. Iannizzotto, S. Mosconi and M. Squassina, Global Hölder regularity for the fractional $p$ -Laplacian, Rev. Mat. Iberoamericana 32(4) (2016) 1353–1392. Crossref, Web of Science, Google Scholar
29. C. Ji and A. Szulkin, A logarithmic Schrödinger equation with asymptotic conditions on the potential, J. Math. Anal. Appl. 437(1) (2016) 241–254. Crossref, Web of Science, Google Scholar
30. N. Laskin, Fractional Quantum Mechanics (World Scientific, Hackensack, NJ, 2018), pp. xv+341. Link, Google Scholar
31. E. H. Lieb and M. Loss, Analysis, Graduate Studies in Mathematics, Vol. 14, 2nd edn. (American Mathematical Society, Providence, RI, 2001), pp. ‘xxii+346. Google Scholar
32. J. Merker, Generalizations of logarithmic Sobolev inequalities, Discrete Contin. Dyn. Syst. S 1 (2008) 329–338. Crossref, Web of Science, Google Scholar
33. G. Molica Bisci, V. Rădulescu and R. Servadei, Variational Methods for Nonlocal Fractional Problems, Vol. 162 (Cambridge University Press, Cambridge, 2016). Crossref, Google Scholar
34. J. Moser, A new proof of De Giorgi’s theorem concerning the regularity problem for elliptic differential equations, Comm. Pure Appl. Math. 13 (1960) 457–468. Crossref, Web of Science, Google Scholar
35. P. Pucci, M. Xiang and B. Zhang, Multiple solutions for nonhomogeneous Schrödinger–Kirchhoff type equations involving the fractional $p$ -Laplacian in $ℝ^{N}$ , Calc. Var. Partial Differential Equations 54(3) (2015) 2785–2806. Crossref, Web of Science, Google Scholar
36. P. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys. 43(2) (1992) 270–291. Crossref, Web of Science, Google Scholar
37. M. Squassina and A. Szulkin, Multiple solutions to logarithmic Schrödinger equations with periodic potential, Calc. Var. Partial Differential Equations 54(1) (2015) 585–597. Crossref, Web of Science, Google Scholar
38. M. Squassina and A. Szulkin, Erratum to: Multiple solutions to logarithmic Schrödinger equations with periodic potential, Calc. Var. Partial Differential Equations 56(3) (2017) 56. Crossref, Web of Science, Google Scholar
39. A. Szulkin, Minimax principles for lower semicontinuous functions and applications to nonlinear boundary value problems, Ann. Inst. H. Poincaré Anal. Non Linéaire 3(2) (1986) 77–109. Crossref, Web of Science, Google Scholar
40. K. Tanaka and C. Zhang, Multi-bump solutions for logarithmic Schrödinger equations, Calc. Var. Partial Differential Equations 56(2) (2017) 33. Crossref, Web of Science, Google Scholar
41. C. E. Torres Ledesma, Existence and symmetry result for fractional p-Laplacian in $ℝ^{N}$ , Commun. Pure Appl. Anal. 16(1) (2017) 99–113. Crossref, Web of Science, Google Scholar
42. L. X. Truong, The Nehari manifold for fractional $p$ -Laplacian equation with logarithmic nonlinearity on whole space, Comput. Math. Appl. 78(12) (2019) 3931–3940. Crossref, Web of Science, Google Scholar
43. M. Willem, Minimax Theorems, Progress in Nonlinear Differential Equations and Their Applications, Vol. 24 (Birkhäuser, Boston, MA, 1996), pp. x+162. Google Scholar
44. M. Xiang, D. Hu and D. Yang, Least energy solutions for fractional Kirchhoff problems with logarithmic nonlinearity, Nonlinear Anal. 198 (2020) 111899. Crossref, Web of Science, Google Scholar
45. K. G. Zloshchastiev, Logarithmic nonlinearity in theories of quantum gravity: Origin of time and observational consequences, Gravit. Cosmol. 16(4) (2010) 288–297. Crossref, Web of Science, Google Scholar