On the Nehari manifold for a logarithmic fractional Schrödinger equation with possibly vanishing potentials

 Access to full text

 Full (PDF)

1. Alves, C. O., Souto, M. A. S., 10.1016/j.jde.2012.01.025, J. Differ. Equations 252 (2012), 5555-5568 9999DOI99999 10.1016/j.jde.2012.01.025 . (2012) Zbl1250.35103MR2902126DOI10.1016/j.jde.2012.01.025
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 9999DOI99999 10.1016/j.jde.2012.11.013 . (2013) Zbl1263.35076MR3003299
3. Ambrosetti, A., Malchiodi, A., 10.1017/CBO9780511618260, Cambridge Studies in Advanced Mathematics 104. Cambridge University Press, Cambridge (2007). (2007) Zbl1125.47052MR2292344DOI10.1017/CBO9780511618260
4. Ambrosetti, A., Wang, Z.-Q., Nonlinear Schrödinger equations with vanishing and decaying potentials, Differ. Integral Equ. 18 (2005), 1321-1332 9999MR99999 2174974 . (2005) Zbl1210.35087MR2174974
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 9999DOI99999 10.1016/j.na.2017.01.006 . (2017) Zbl1368.35242MR3631741
6. Benci, V., Grisanti, C. R., Micheletti, A. M., Existence of solutions for the nonlinear Schrödinger equation with $V\left(\infty \right)=0$, Contributions to Nonlinear Analysis Progress in Nonlinear Differential Equations and Their Applications 66. Birkhäuser, Basel (2006), 53-65 9999DOI99999 10.1007/3-7643-7401-24 . (2006) Zbl1231.35225MR2187794
7. Berestycki, H., Lions, P.-L., Nonlinear scalar field equations. I: Existence of a ground state, Arch. Ration. Mech. Anal. 82 (1983), 313-345 9999DOI99999 10.1007/BF00250555 . (1983) Zbl0533.35029MR0695535
8. Bia{ł}ynicki-Birula, I., Mycielski, J., Nonlinear wave mechanics, Ann. Phys. 100 (1976), 62-93 9999DOI99999 10.1016/0003-4916(76)90057-9 . (1976) MR0426670
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 9999DOI99999 10.1016/S0022-0396(03)00121-9 . (2003) Zbl1074.35032MR1998965
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 9999DOI99999 10.1007/978-3-642-25361-43 . (2012) Zbl1266.35060MR3289358
11. Campa, I., Degiovanni, M., Subdifferential calculus and nonsmooth critical point theory, SIAM J. Optim. 10 (2000), 1020-1048 9999DOI99999 10.1137/S1052623499353169 . (2000) Zbl1042.49018MR1777078
12. Cazenave, T., 10.1016/0362-546X(83)90022-6, Nonlinear Anal., Theory Methods Appl. 7 (1983), 1127-1140. (1983) Zbl0529.35068MR0719365DOI10.1016/0362-546X(83)90022-6
13. Cazenave, T., Haraux, A., Équations d'évolution avec non linéarité logarithmique, Ann. Fac. Sci. Toulouse, Math. (5) 2 (1980), 21-51 French 9999DOI99999 10.5802/afst.543 . (1980) Zbl0411.35051MR0583902
14. Chang, X., Ground state solutions of asymptotically linear fractional Schrödinger equations, J. Math. Phys. 54 (2013), Article ID 061504, 10 pages 9999DOI99999 10.1063/1.4809933 . (2013) Zbl1282.81072MR3112523
15. Chen, W., Deng, S., 10.1007/s00033-014-0486-6, Z. Angew. Math. Phys. 66 (2015), 1387-1400. (2015) Zbl1321.35253MR3377693DOI10.1007/s00033-014-0486-6
16. Cheng, M., Bound state for the fractional Schrödinger equation with unbounded potential, J. Math. Phys. 53 (2012), Article ID 043507, 7 pages 9999DOI99999 10.1063/1.3701574 . (2012) Zbl1275.81030MR2953151
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 9999DOI99999 10.12775/TMNA.1993.012 . (1993) Zbl0789.58021MR1215263
18. D'Avenia, P., Montefusco, E., Squassina, M., On the logarithmic Schrödinger equation, Commun. Contemp. Math. 16 (2014), Article ID 1350032, 15 pages 9999DOI99999 10.1142/S0219199713500326 . (2014) Zbl1292.35259MR3195154
19. Degiovanni, M., Zani, S., 10.1016/S0895-7177(00)00211-9, Math. Comput. Modelling 32 (2000), 1377-1393. (2000) Zbl0970.35038MR1800662DOI10.1016/S0895-7177(00)00211-9
20. Nezza, E. Di, Palatucci, G., Valdinoci, E., Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math. 136 (2012), 521-573 9999DOI99999 10.1016/j.bulsci.2011.12.004 . (2012) Zbl1252.46023MR2944369
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 9999DOI99999 10.1017/S0308210500023787 . (1997) Zbl0880.35045MR1465416
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 9999DOI99999 10.1515/ans-2008-0207 . (2008) Zbl1168.35433MR2402826
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 9999DOI99999 10.1103/PhysRevA.32.1201 . (1985) 
24. Ji, C., Szulkin, A., 10.1016/j.jmaa.2015.11.071, J. Math. Anal. Appl. 437 (2016), 241-254. (2016) Zbl1333.35010MR3451965DOI10.1016/j.jmaa.2015.11.071
25. Khoutir, S., Chen, H., Existence of infinitely many high energy solutions for a fractional Schrödinger equation in ${ℝ}^N$, Appl. Math. Lett. 61 (2016), 156-162 9999DOI99999 10.1016/j.aml.2016.06.001 . (2016) Zbl1386.35444MR3518463
26. Laskin, N., Fractional quantum mechanics and Lévy path integrals, Phys. Lett., A 268 (2000), 298-305 9999DOI99999 10.1016/S0375-9601(00)00201-2 . (2000) Zbl0948.81595MR1755089
27. Laskin, N., Fractional Schrödinger equation, Phys. Rev. E (3) 66 (2002), Article ID 056108, 7 pages 9999DOI99999 10.1103/PhysRevE.66.056108 . (2002) MR1948569
28. Perera, K., Squassina, M., Yang, Y., Critical fractional $p$-Laplacian problems with possibly vanishing potentials, J. Math. Anal. Appl. 433 (2016), 818-831 9999DOI99999 10.1016/j.jmaa.2015.08.024 . (2016) Zbl1403.35319MR3398738
29. Secchi, S., Ground state solutions for nonlinear fractional Schrödinger equations in ${ℝ}^N$, J. Math. Phys. 54 (2013), Article ID 031501, 17 pages 9999DOI99999 10.1063/1.4793990 . (2013) Zbl1281.81034MR3059423
30. Shang, X., Zhang, J., Ground states for fractional Schrödinger equations with critical growth, Nonlinearity 27 (2014), 187-207 9999DOI99999 10.1088/0951-7715/27/2/187 . (2014) Zbl1287.35027MR3153832
31. Shang, X., Zhang, J., Yang, Y., 10.1063/1.4835355, J. Math. Phys. 54 (2013), Article ID 121502, 20 pages. (2013) Zbl1290.35251MR3156081DOI10.1063/1.4835355
32. Squassina, M., Szulkin, A., 10.1007/s00526-014-0796-8, Calc. Var. Partial Differ. Equ. 54 (2015), 585-597. (2015) Zbl1326.35358MR3385171DOI10.1007/s00526-014-0796-8
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 9999DOI99999 10.1016/S0294-1449(16)30389-4 . (1986) Zbl0612.58011MR0837231
34. Teng, K., Multiple solutions for a class of fractional Schrödinger equations in ${ℝ}^N$, Nonlinear Anal., Real World Appl. 21 (2015), 76-86 9999DOI99999 10.1016/j.nonrwa.2014.06.008 . (2015) Zbl1302.35415MR3261580
35. Ledesma, C. E. Torres, 10.3934/cpaa.2017004, Commun. Pure Appl. Anal. (2017), 16 99-113. (2017) Zbl1364.35426MR3583517DOI10.3934/cpaa.2017004
36. Zloshchastiev, K. G., 10.1134/S0202289310040067, Grav. Cosmol. 16 (2010), 288-297. (2010) Zbl1232.83044MR2740900DOI10.1134/S0202289310040067