On nonlinear fractional Schrödinger equations with indefinite and Hardy potentials. (English) Zbl 1522.35474
Summary: This paper is concerned with a class of fractional Schrödinger equation with Hardy potential
\[
(-\Delta)^s u + V(x) u - \frac{\kappa}{|x|^{2s}} u = f(x, u), \quad x \in \mathbb{R}^N,
\]
where \(s \in (0, 1)\) and \(\kappa \geqslant 0\) is a parameter. Under some suitable conditions on the potential \(V\) and the nonlinearity \(f\), we prove the existence of ground state solutions when the parameter \(\kappa\) lies in a given range by using the non-Nehari manifold method. Moreover, we investigate the continuous dependence of ground state energy about \(\kappa\). Finally, we are able to explore the asymptotic behavior of ground state solutions when \(\kappa\) tends to 0.
MSC:
| 35Q55 | NLS equations (nonlinear Schrödinger equations) |
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
| 35A02 | Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness |
| 26A33 | Fractional derivatives and integrals |
| 35R11 | Fractional partial differential equations |
| 35R10 | Partial functional-differential equations |
| 35R02 | PDEs on graphs and networks (ramified or polygonal spaces) |
Keywords:
fractional Schrödinger equation; Hardy potential; strongly indefinite functional; ground state solutionsReferences:
| [1] | F. Bernini, B. Bieganowski and S. Secchi, Semirelativistic Choquard equations with singular potentials and general non-linearities arising from Hartree-Fock theory, Nonlinear Analysis 217 (2022), 112738. doi:10.1016/j.na.2021.112738. · Zbl 1484.35344 · doi:10.1016/j.na.2021.112738 |
| [2] | B. Bieganowski, The fractional Schrödinger equation with Hardy-type potentials and sign-changing nonlinearities, Non-linear Analysis 176 (2018), 117-140. doi:10.1016/j.na.2018.06.009. · Zbl 1403.35270 · doi:10.1016/j.na.2018.06.009 |
| [3] | X. Cabré and Y. Sire, Nonlinear equations for fractional Laplacians, I: Regularity, maximum principles, and Hamiltonian estimates, Ann. Inst. H. Poincaré Anal. Non Linéaire 31 (2014), 23-53. doi:10.1016/j.anihpc.2013.02.001. · Zbl 1286.35248 · doi:10.1016/j.anihpc.2013.02.001 |
| [4] | L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equa-tions 32 (2007), 1245-1260. doi:10.1080/03605300600987306. · Zbl 1143.26002 · doi:10.1080/03605300600987306 |
| [5] | D. Cao and P. Han, Solutions for semilinear elliptic equations with critical exponents and Hardy potential, J. Differential Equations 205 (2004), 521-537. doi:10.1016/j.jde.2004.03.005. · Zbl 1154.35346 · doi:10.1016/j.jde.2004.03.005 |
| [6] | D. Cao and S. Yan, Infinitely many solutions for an elliptic problem involving critical Sobolev growth and Hardy potential, Calc. Var. Partial Differential Equations 38 (2010), 471-501. doi:10.1007/s00526-009-0295-5. · Zbl 1194.35161 · doi:10.1007/s00526-009-0295-5 |
| [7] | X. Chang and Z. Wang, Ground state of scalar field equations involving a fractional Laplacian with general nonlinearity, Nonlinearity 26 (2013), 479-494. doi:10.1088/0951-7715/26/2/479. · Zbl 1276.35080 · doi:10.1088/0951-7715/26/2/479 |
| [8] | P. Chen, X. Tang and L. Zhang, Non-Nehari manifold method for Hamiltonian elliptic system with Hardy potential: Existence and asymptotic properties of ground state solution, J. Geom. Anal. 32 (2022), 46. doi:10.1007/s12220-021-00739-5. · Zbl 1484.35185 · doi:10.1007/s12220-021-00739-5 |
| [9] | S. Chen, X. Tang and J. Wei, Nehari-type ground state solutions for a Choquard equation with doubly critical exponents, Adv. Nonlinear Anal. 10 (2021), 152-171. doi:10.1515/anona-2020-0118. · Zbl 1440.35130 · doi:10.1515/anona-2020-0118 |
| [10] | Z. Chen and W. Zou, On an elliptic problem with critical exponent and Hardy potential, J. Differential Equations 252 (2012), 969-987. doi:10.1016/j.jde.2011.09.042. · Zbl 1233.35078 · doi:10.1016/j.jde.2011.09.042 |
| [11] | Y. Ding, Variational Methods for Strongly Indefinite Problems, World Scientific Press, 2008. |
| [12] | M.M. Fall and V. Felli, Sharp essential self-adjointness of relativistic Schrödinger operators with a singular potential, J. Functional Analysis 267 (2014), 1851-1877. doi:10.1016/j.jfa.2014.06.010. · Zbl 1295.35377 · doi:10.1016/j.jfa.2014.06.010 |
| [13] | M.M. Fall and V. Felli, Unique continuation properties for relativistic Schrödinger operators with a singular potential, Discrete Contin. Dyn. Syst. 35 (2015), 5827-5867. doi:10.3934/dcds.2015.35.5827. · Zbl 1336.35356 · doi:10.3934/dcds.2015.35.5827 |
| [14] | F. Fang and C. Ji, On a fractional Schrödinger equation with periodic potential, Comput. Math. Appl. 78 (2019), 1517-1530. doi:10.1016/j.camwa.2019.03.044. · Zbl 1442.35508 · doi:10.1016/j.camwa.2019.03.044 |
| [15] | V. Felli, E. Marchini and S. Terracini, On Schrödinger operators with multipolar inverse-square potentials, J. Funct. Anal. 250 (2007), 265-316. doi:10.1016/j.jfa.2006.10.019. · Zbl 1222.35074 · doi:10.1016/j.jfa.2006.10.019 |
| [16] | V. Felli and S. Terracini, Elliptic equations with multi-singular inverse-square potentials and critical nonlinearity, Comm. Partial Differential Equations 31 (2006), 469-495. doi:10.1080/03605300500394439. · Zbl 1206.35104 · doi:10.1080/03605300500394439 |
| [17] | P. Felmer, A. Quaas and J. Tan, Positive solutions of nonlinear Schrödinger equation with the fractional Laplacian, Proc. Roy. Soc. Edinburgh Sect. A 142 (2012), 1237-1262. doi:10.1017/S0308210511000746. · Zbl 1290.35308 · doi:10.1017/S0308210511000746 |
| [18] | R.L. Frank and R. Seiringer, Nonlinear ground state representations and sharp Hardy inequalities, J. Funct. Anal. 255 (2008), 3407-3430. doi:10.1016/j.jfa.2008.05.015. · Zbl 1189.26031 · doi:10.1016/j.jfa.2008.05.015 |
| [19] | W.M. Frank, D.J. Land and R.M. Spector, Singular potentials, Rev. Modern Phys. 43(1) (1971), 36-98. doi:10.1103/ RevModPhys.43.36. · doi:10.1103/RevModPhys.43.36 |
| [20] | Q. Guo and J. Mederski, Ground states of nonlinear Schrödinger equations with sum of periodic and inverse-square potentials, J. Differential Equations 260 (2016), 4180-4202. doi:10.1016/j.jde.2015.11.006. · Zbl 1335.35232 · doi:10.1016/j.jde.2015.11.006 |
| [21] | W. Kryszewski and A. Szulkin, Generalized linking theorem with an application to semilinear Schrödinger equation, Adv. Differ. Equat. 3 (1998), 441-472. · Zbl 0947.35061 |
| [22] | N. Laskin, Fractional Schrödinger equations, Phys. Rev. E 66 (2002), 056108. doi:10.1103/PhysRevE.66.056108. · doi:10.1103/PhysRevE.66.056108 |
| [23] | P.L. Lions, The concentration compactness principle in the calculus of variations. The locally compact case. Part II, Ann. Inst. H. Poincaré, Anal. Non Linéaire 1 (1984), 223-283. doi:10.1016/s0294-1449(16)30422-x. · Zbl 0704.49004 · doi:10.1016/s0294-1449(16)30422-x |
| [24] | G. Molica Bisci and V. Rȃdulescu, Ground state solutions of scalar field fractional Schrödinger equations, Calc. Var. Partial Differential Equations 54 (2015), 2985-3008. doi:10.1007/s00526-015-0891-5. · Zbl 1330.35495 · doi:10.1007/s00526-015-0891-5 |
| [25] | G. Molica Bisci, V.D. Rȃdulescu and R. Servadei, Variational Methods for Nonlocal Fractional Problems, Encyclopedia of Mathematics and Its Applications, Vol. 162, Cambridge University Press, Cambridge, 2016. · Zbl 1356.49003 |
| [26] | A. Pankov, Periodic nonlinear Schrödinger equation with application to photonic crystals, Milan J. Math. 73 (2005), 259-287. doi:10.1007/s00032-005-0047-8. · Zbl 1225.35222 · doi:10.1007/s00032-005-0047-8 |
| [27] | Y. Pu, J. Liu and C.L. Tang, Existence of weak solutions for a class of fractional Schrödinger equations with periodic potential, Comput. Math. Appl. 73 (2017), 465-482. doi:10.1016/j.camwa.2016.12.004. · Zbl 1524.35132 · doi:10.1016/j.camwa.2016.12.004 |
| [28] | D. Ruiz and M. Willem, Elliptic problems with critical exponents and Hardy potentials, J. Differential Equations 190 (2003), 524-538. doi:10.1016/S0022-0396(02)00178-X. · Zbl 1163.35383 · doi:10.1016/S0022-0396(02)00178-X |
| [29] | E.A. Silva and G.F. Vieira, Quasilinear asymptotically periodic Schrödinger equations with critical growth, Calc. Var. PDE 39 (2010), 1-33. doi:10.1007/s00526-009-0299-1. · Zbl 1223.35290 · doi:10.1007/s00526-009-0299-1 |
| [30] | A. Szulkin and T. Weth, Ground state solutions for some indefinite variational problems, J. Funct. Anal. 257 (2009), 3802-3822. doi:10.1016/j.jfa.2009.09.013. · Zbl 1178.35352 · doi:10.1016/j.jfa.2009.09.013 |
| [31] | X. Tang, Non-Nehari manifold method for asymptotically periodic Schrödinger equations, Sci. China Math. 58 (2015), 715-728. doi:10.1007/s11425-014-4957-1. · Zbl 1321.35055 · doi:10.1007/s11425-014-4957-1 |
| [32] | M. Willem, Minimax Theorems, Birkhäuser, Berlin, 1996. · Zbl 0856.49001 |
| [33] | Z. Yang and F. Zhao, Multiplicity and concentration behaviour of solutions for a fractional Choquard equation with critical growth, Adv. Nonlinear Anal. 10 (2021), 732-774. doi:10.1515/anona-2020-0151. · Zbl 1466.35304 · doi:10.1515/anona-2020-0151 |
| [34] | H. Zhang, J. Xu and F. Zhang, Existence and multiplicity of solutions for superlinear fractional Schrödinger equations in R N , J. Math. Phys. 56 (2015), 091502. doi:10.1063/1.4929660. · Zbl 1328.35225 · doi:10.1063/1.4929660 |
| [35] | J. Zhang and W. Zhang, Existence and asymptotic behavior of ground states for Schrödinger systems with Hardy potential, Nonlinear Analysis 189 (2019), 111586. doi:10.1016/j.na.2019.111586. · Zbl 1427.35037 · doi:10.1016/j.na.2019.111586 |
| [36] | J. Zhang, W. Zhang and X. Tang, Ground state solutions for Hamiltonian elliptic system with inverse square potential, Discrete Contin. Dyn. Syst. 37 (2017), 4565-4583. doi:10.3934/dcds.2017195. · Zbl 1370.35111 · doi:10.3934/dcds.2017195 |
| [37] | W. Zhang, S. Yuan and L. Wen, Existence and concentration of ground-states for fractional Choquard equation with indefinite potential, Adv. Nonlinear Anal. 11 (2022), 1152-1578. doi:10.1515/anona-2022-0255. · Zbl 1494.35173 · doi:10.1515/anona-2022-0255 |
| [38] | W. Zhang, J. Zhang and H. Mi, On fractional Schrödinger equation with periodic and asymptotically periodic conditions, Compu. Math. Appl. 74 (2017), 1321-1332. doi:10.1016/j.camwa.2017.06.017. · Zbl 1394.35572 · doi:10.1016/j.camwa.2017.06.017 |
| [39] | W. Zhang, J. Zhang and H. Mi, Ground states and multiple solutions for Hamiltonian elliptic system with gradient term, Adv. Nonlinear Anal. 10 (2021), 331-352. doi:10.1515/anona-2020-0113. · Zbl 1448.35178 · doi:10.1515/anona-2020-0113 |
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