| Title:
|
On the Nehari manifold for a logarithmic fractional Schrödinger equation with possibly vanishing potentials (English) |
| Author:
|
Le, Cong Nhan |
| Author:
|
Le, Xuan Truong |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
147 |
| Issue:
|
1 |
| Year:
|
2022 |
| Pages:
|
33-49 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We study a class of logarithmic fractional Schrödinger equations with possibly vanishing potentials. By using the fibrering maps and the Nehari manifold we obtain the existence of at least one nontrivial solution. (English) |
| Keyword:
|
Nehari manifold |
| Keyword:
|
fibrering maps |
| Keyword:
|
vanishing potential |
| Keyword:
|
logarithmic nonlinearity |
| MSC:
|
35J60 |
| MSC:
|
47J30 |
| idZBL:
|
Zbl 07547240 |
| idMR:
|
MR4387467 |
| DOI:
|
10.21136/MB.2021.0143-19 |
| . |
| Date available:
|
2022-04-17T18:08:22Z |
| Last updated:
|
2022-09-06 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/149588 |
| . |
| Reference:
|
[1] Alves, C. O., Souto, M. A. S.: Existence of solutions for a class of elliptic equations in $\mathbb R^N$ with vanishing potentials.J. Differ. Equations 252 (2012), 5555-5568 \99999DOI99999 10.1016/j.jde.2012.01.025 . Zbl 1250.35103, MR 2902126, 10.1016/j.jde.2012.01.025 |
| Reference:
|
[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. Equations 254 (2013), 1977-1991 \99999DOI99999 10.1016/j.jde.2012.11.013 . Zbl 1263.35076, MR 3003299 |
| Reference:
|
[3] Ambrosetti, A., Malchiodi, A.: Nonlinear Analysis and Semilinear Elliptic Problems.Cambridge Studies in Advanced Mathematics 104. Cambridge University Press, Cambridge (2007). Zbl 1125.47052, MR 2292344, 10.1017/CBO9780511618260 |
| Reference:
|
[4] Ambrosetti, A., Wang, Z.-Q.: Nonlinear Schrödinger equations with vanishing and decaying potentials.Differ. Integral Equ. 18 (2005), 1321-1332 \99999MR99999 2174974 . Zbl 1210.35087, MR 2174974 |
| Reference:
|
[5] Ardila, A. H.: Existence and stability of standing waves for nonlinear fractional Schrödinger equation with logarithmic nonlinearity.Nonlinear Anal., Theory Methods Appl. 155 (2017), 52-64 \99999DOI99999 10.1016/j.na.2017.01.006 . Zbl 1368.35242, MR 3631741 |
| Reference:
|
[6] Benci, V., Grisanti, C. R., Micheletti, A. M.: Existence of solutions for the nonlinear Schrödinger equation with $V(\infty)=0$.Contributions to Nonlinear Analysis Progress in Nonlinear Differential Equations and Their Applications 66. Birkhäuser, Basel (2006), 53-65 \99999DOI99999 10.1007/3-7643-7401-2_4 . Zbl 1231.35225, MR 2187794 |
| Reference:
|
[7] Berestycki, H., Lions, P.-L.: Nonlinear scalar field equations. I: Existence of a ground state.Arch. Ration. Mech. Anal. 82 (1983), 313-345 \99999DOI99999 10.1007/BF00250555 . Zbl 0533.35029, MR 0695535 |
| Reference:
|
[8] Bia{ł}ynicki-Birula, I., Mycielski, J.: Nonlinear wave mechanics.Ann. Phys. 100 (1976), 62-93 \99999DOI99999 10.1016/0003-4916(76)90057-9 . MR 0426670 |
| Reference:
|
[9] Brown, K. J., Zhang, Y.: The Nehari manifold for a semilinear elliptic problem with a sign-changing weight function.J. Differ. Equations 193 (2003), 481-499 \99999DOI99999 10.1016/S0022-0396(03)00121-9 . Zbl 1074.35032, MR 1998965 |
| Reference:
|
[10] Caffarelli, L.: Non-local diffusions, drifts and games.Nonlinear Partial Differential Equations: The Abel symposium 2010 Abel Symposia 7. Springer, Berlin (2012), 37-52 \99999DOI99999 10.1007/978-3-642-25361-4_3 . Zbl 1266.35060, MR 3289358 |
| Reference:
|
[11] Campa, I., Degiovanni, M.: Subdifferential calculus and nonsmooth critical point theory.SIAM J. Optim. 10 (2000), 1020-1048 \99999DOI99999 10.1137/S1052623499353169 . Zbl 1042.49018, MR 1777078 |
| Reference:
|
[12] Cazenave, T.: Stable solutions of the logarithmic Schrödinger equation.Nonlinear Anal., Theory Methods Appl. 7 (1983), 1127-1140. Zbl 0529.35068, MR 0719365, 10.1016/0362-546X(83)90022-6 |
| Reference:
|
[13] Cazenave, T., Haraux, A.: Équations d'évolution avec non linéarité logarithmique.Ann. Fac. Sci. Toulouse, Math. (5) 2 (1980), 21-51 French \99999DOI99999 10.5802/afst.543 . Zbl 0411.35051, MR 0583902 |
| Reference:
|
[14] Chang, X.: Ground state solutions of asymptotically linear fractional Schrödinger equations.J. Math. Phys. 54 (2013), Article ID 061504, 10 pages \99999DOI99999 10.1063/1.4809933 . Zbl 1282.81072, MR 3112523 |
| Reference:
|
[15] Chen, W., Deng, S.: The Nehari manifold for nonlocal elliptic operators involving concave-convex nonlinearities.Z. Angew. Math. Phys. 66 (2015), 1387-1400. Zbl 1321.35253, MR 3377693, 10.1007/s00033-014-0486-6 |
| Reference:
|
[16] Cheng, M.: Bound state for the fractional Schrödinger equation with unbounded potential.J. Math. Phys. 53 (2012), Article ID 043507, 7 pages \99999DOI99999 10.1063/1.3701574 . Zbl 1275.81030, MR 2953151 |
| Reference:
|
[17] Corvellec, J.-N., Degiovanni, M., Marzocchi, M.: Deformation properties for continuous functionals and critical point theory.Topol. Methods Nonlinear Anal. 1 (1993), 151-171 \99999DOI99999 10.12775/TMNA.1993.012 . Zbl 0789.58021, MR 1215263 |
| Reference:
|
[18] D'Avenia, P., Montefusco, E., Squassina, M.: On the logarithmic Schrödinger equation.Commun. Contemp. Math. 16 (2014), Article ID 1350032, 15 pages \99999DOI99999 10.1142/S0219199713500326 . Zbl 1292.35259, MR 3195154 |
| Reference:
|
[19] Degiovanni, M., Zani, S.: Multiple solutions of semilinear elliptic equations with one-sided growth conditions.Math. Comput. Modelling 32 (2000), 1377-1393. Zbl 0970.35038, MR 1800662, 10.1016/S0895-7177(00)00211-9 |
| Reference:
|
[20] Nezza, E. Di, Palatucci, G., Valdinoci, E.: Hitchhiker's guide to the fractional Sobolev spaces.Bull. Sci. Math. 136 (2012), 521-573 \99999DOI99999 10.1016/j.bulsci.2011.12.004 . Zbl 1252.46023, MR 2944369 |
| Reference:
|
[21] Drábek, P., Pohozaev, S. I.: Positive solutions for the $p$-Laplacian: Application of the fibrering method.Proc. R. Soc. Edinb., Sect. A 127 (1997), 703-726 \99999DOI99999 10.1017/S0308210500023787 . Zbl 0880.35045, MR 1465416 |
| Reference:
|
[22] Furtado, M. F., Maia, L. A., Medeiros, E. S.: Positive and nodal solutions for a nonlinear Schrödinger equation with indefinite potential.Adv. Nonlinear Stud. 8 (2008), 353-373 \99999DOI99999 10.1515/ans-2008-0207 . Zbl 1168.35433, MR 2402826 |
| Reference:
|
[23] Hefter, E. F.: Application of the nonlinear Schrödinger equation with a logarithmic inhomogeneous term to nuclear physics.Phys. Rev. 32(A) (1985), 1201-1204 \99999DOI99999 10.1103/PhysRevA.32.1201 . |
| Reference:
|
[24] Ji, C., Szulkin, A.: A logarithmic Schrödinger equation with asymptotic conditions on the potential.J. Math. Anal. Appl. 437 (2016), 241-254. Zbl 1333.35010, MR 3451965, 10.1016/j.jmaa.2015.11.071 |
| Reference:
|
[25] Khoutir, S., Chen, H.: Existence of infinitely many high energy solutions for a fractional Schrödinger equation in $\mathbb R^N$.Appl. Math. Lett. 61 (2016), 156-162 \99999DOI99999 10.1016/j.aml.2016.06.001 . Zbl 1386.35444, MR 3518463 |
| Reference:
|
[26] Laskin, N.: Fractional quantum mechanics and Lévy path integrals.Phys. Lett., A 268 (2000), 298-305 \99999DOI99999 10.1016/S0375-9601(00)00201-2 . Zbl 0948.81595, MR 1755089 |
| Reference:
|
[27] Laskin, N.: Fractional Schrödinger equation.Phys. Rev. E (3) 66 (2002), Article ID 056108, 7 pages \99999DOI99999 10.1103/PhysRevE.66.056108 . MR 1948569 |
| Reference:
|
[28] Perera, K., Squassina, M., Yang, Y.: Critical fractional $p$-Laplacian problems with possibly vanishing potentials.J. Math. Anal. Appl. 433 (2016), 818-831 \99999DOI99999 10.1016/j.jmaa.2015.08.024 . Zbl 1403.35319, MR 3398738 |
| Reference:
|
[29] Secchi, S.: Ground state solutions for nonlinear fractional Schrödinger equations in $\mathbb R^N$.J. Math. Phys. 54 (2013), Article ID 031501, 17 pages \99999DOI99999 10.1063/1.4793990 . Zbl 1281.81034, MR 3059423 |
| Reference:
|
[30] Shang, X., Zhang, J.: Ground states for fractional Schrödinger equations with critical growth.Nonlinearity 27 (2014), 187-207 \99999DOI99999 10.1088/0951-7715/27/2/187 . Zbl 1287.35027, MR 3153832 |
| Reference:
|
[31] Shang, X., Zhang, J., Yang, Y.: On fractional Schrödinger equation in $\mathbb R^N$ with critical growth.J. Math. Phys. 54 (2013), Article ID 121502, 20 pages. Zbl 1290.35251, MR 3156081, 10.1063/1.4835355 |
| Reference:
|
[32] Squassina, M., Szulkin, A.: Multiple solutions to logarithmic Schrödinger equations with periodic potential.Calc. Var. Partial Differ. Equ. 54 (2015), 585-597. Zbl 1326.35358, MR 3385171, 10.1007/s00526-014-0796-8 |
| Reference:
|
[33] Szulkin, A.: Minimax principles for lower semicontinuous functions and applications to nonlinear boundary value problems.Ann. Inst. Henri Poincaré, Anal. Non Linéaire 3 (1986), 77-109 \99999DOI99999 10.1016/S0294-1449(16)30389-4 . Zbl 0612.58011, MR 0837231 |
| Reference:
|
[34] Teng, K.: Multiple solutions for a class of fractional Schrödinger equations in $\mathbb R^N$.Nonlinear Anal., Real World Appl. 21 (2015), 76-86 \99999DOI99999 10.1016/j.nonrwa.2014.06.008 . Zbl 1302.35415, MR 3261580 |
| Reference:
|
[35] Ledesma, C. E. Torres: Existence and symmetry result for fractional $p$-Laplacian in $\Bbb{R}^n$.Commun. Pure Appl. Anal. (2017), 16 99-113. Zbl 1364.35426, MR 3583517, 10.3934/cpaa.2017004 |
| Reference:
|
[36] Zloshchastiev, K. G.: Logarithmic nonlinearity in theories of quantum gravity: Origin of time and observational consequences.Grav. Cosmol. 16 (2010), 288-297. Zbl 1232.83044, MR 2740900, 10.1134/S0202289310040067 |
| . |
