Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-23T08:27:02.307Z Has data issue: false hasContentIssue false
  • English
  • Français

Ground States of some Fractional Schrödinger Equations on ℝN

Part of: Elliptic equations and systems Equations and inequalities involving nonlinear operators

Published online by Cambridge University Press:  27 October 2014

Xiaojun Chang
Xiaojun Chang*
Affiliation:
School of Mathematics and Statistics, Northeast Normal University, Changchun 130024, Jilin, People's Republic of China College of Mathematics, Jilin University, Changchun 130012, Jilin, People's Republic of China, ([email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper, we study a time-independent fractional Schrödinger equation of the form (−Δ)su + V(x)u = g(u) in ℝN, where N ≥, s ∈ (0,1) and (−Δ)s is the fractional Laplacian. By variational methods, we prove the existence of ground state solutions when V is unbounded and the nonlinearity g is subcritical and satisfies the following geometry condition:

Keywords

ground statesfractional Schrödinger equationvariational methods

MSC classification

Primary: 35J60: Nonlinear elliptic equations
Secondary: 47J30: Variational methods
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 58 , Issue 2 , June 2015 , pp. 305 - 321
Copyright
Copyright © Edinburgh Mathematical Society 2012 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Ambrosetti, A. and Rabinowitz, P. H., Dual variational methods in critical point theory and applications, J. Funct. Analysis 14 (1973), 349–381.CrossRefGoogle Scholar
2.Ambrosetti, A., Malchiodi, A. and Secchi, S., Multiplicity results for some nonlinear Schrödinger equations with potentials, Arch. Ration. Mech. Analysis 159 (2001), 253–271.CrossRefGoogle Scholar
3.Azzollini, A. and Pomponio, A., On the Schrödinger equation in ℝN under the effect of a general nonlinear term, Indiana Univ. Math. J. 58 (2009), 1361–1378.CrossRefGoogle Scholar
4.Barrios, B., Colorado, E., Depablo, A. and Sánchez, U., On some critical problems for the fractional Laplacian operator, J. Diff. Eqns 252 (2012), 6133–6162.CrossRefGoogle Scholar
5.Bartsch, T. and Wang, Z.-Q., Existence and multiplicity results for some superlinear elliptic problems on ℝN, Commun. PDEs 20 (1995), 1725–1741.CrossRefGoogle Scholar
6.Bartsch, T., Wang, Z.-Q. and Willem, M., The Dirichlet problem for superlinear elliptic equations, in Handbook of differential equations: stationary partial differential equations, Chapter 1, Volume 2, pp. 1–55 (Elsevier, 2005).CrossRefGoogle Scholar
7.Brändle, C., Colorado, E. and Depablo, A., A concave–convex elliptic problem involving the fractional Laplacian, Proc. R. Soc. Edinb. A 143 (2013), 39–71.CrossRefGoogle Scholar
8.Brezis, H., Functional analysis, Sobolev spaces and partial differential equations (Springer, 2011).CrossRefGoogle Scholar
9.Cabré, X. and Sire, Y., Nonlinear equations for fractional Laplacians I: regularity, maximum principle and Hamiltonian estimates, Annales Inst. H. Poincaré Analyse Non Linéaire 31(1) (2013), 23–53.CrossRefGoogle Scholar
10.Cabré, X. and Tan, J. G., Positive solutions of nonlinear problems involving the square root of the Laplacian, Adv. Math. 224 (2010), 2052–2093.CrossRefGoogle Scholar
11.Caffarelli, L. and Silvestre, L., An extension problem related to the fractional Laplacian, Commun. PDEs 32 (2007), 1245–1260.CrossRefGoogle Scholar
12.Caffarelli, L., Salsa, S. and Silvestre, L., Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian, Invent. Math. 171 (2008), 425–461.CrossRefGoogle Scholar
13.Capella, A., Dávila, J., Dupaigne, L. and Sire, Y., Regularity of radial extremal solutions for some nonlocal semilinear equations, Commun. PDEs 36 (2011), 1353–1384.CrossRefGoogle Scholar
14.Chang, X. J., Ground state solutions of asymptotically linear fractional Schrödinger equations, J. Math. Phys. 54 (2013), 061504.CrossRefGoogle Scholar
15.Chang, X. J. and Wang, Z.-Q., Ground state of scalar field equations involving fractional Laplacian with general nonlinearity, Nonlinearity 26 (2013), 479–494.CrossRefGoogle Scholar
16.Chang, X. J. and Wang, Z.-Q., Nodal and multiple solutions of nonlinear problems involving the fractional Laplacian, J. Diff. Eqns 256(8) (2014), 2965–2992.CrossRefGoogle Scholar
17.Cho, Y., Hwang, G., Kwon, S. and Lee, S., Profile decompositions and blowup phenomena of mass critical fractional Schröodinger equations, Nonlin. Analysis 86 (2013), 12–29.CrossRefGoogle Scholar
18.Zelati, V. Coti and Nolasco, M., Existence of ground states for nonlinear, pseudo-relativistic Schroödinger equations, Rend. Lincei Mat. Appl. 22 (2011), 51–72.Google Scholar
19.Zelati, V. Coti and Nolasco, M., Ground states for pseudo-relativistic Hartree equations of critical type, Rev. Mat. Ibero. 29 (2013), 1421–1436.CrossRefGoogle Scholar
20.Dipierro, S., Palatucci, G. and Valdinoci, E., Existence and symmetry results for a Schröodinger type problem involving the fractional Laplacian, Matematiche 68 (2013), 201–216.Google Scholar
21.Fabes, E. B., Kenig, C. E. and Serapioni, R. P., The local regularity of solutions of degenerate elliptic equations, Commun. PDEs 7 (1982), 77–116.CrossRefGoogle Scholar
22.Fall, M. M. and Valdinoci, E., Uniqueness and nondegeneracy of positive solutions of (−Δ)su + u = u p in ℝN when s is close to 1, Commun. Math. Phys. 329 (2014), 383–404.CrossRefGoogle Scholar
23.Felmer, P., Quaas, A. and Tan, J. G., Positive solutions of nonlinear Schródinger equation with the fractional Laplacian, Proc. R. Soc. Edinb. A 142 (2012), 1237–1262.CrossRefGoogle Scholar
24.Frank, R. and Lenzmann, E., Uniqueness and nondegeneracy of ground states for (−Δ)sQ + Q − Q α+1 = 0 in ℝ, Acta. Math. 201 (2013), 261–318.CrossRefGoogle Scholar
25.Frank, R. L., Lenzmann, E. and Silvestre, L., Uniqueness of radial solutions for the fractional Laplacians in R, preprint (arXiv:1302.2652, 2013).Google Scholar
26.Guo, B. L. and Huo, Z.H., Global well-posedness for the fractional nonlinear Schrödinger equation, Commun. PDEs 36 (2011), 1245–1260.Google Scholar
27.Jeanjean, L., On the existence of bounded Palais–Smale sequences and application to a Landesman-Lazer type problem set on ℝN, Proc. R. Soc. Edinb. A 129 (1999), 787–809.CrossRefGoogle Scholar
28.Laskin, N., Fractional quantum mechanics and Lévy path integrals, Phys. Lett. A 268 (2000), 298–305.CrossRefGoogle Scholar
29.Laskin, N., Fractional Schrödinger equation, Phys. Rev. E 66 (2002), 056108.Google ScholarPubMed
30.Li, Y. Q., Wang, Z.-Q. and Zeng, J., Ground states of nonlinear Schrödinger equations with potentials, Annales Inst. H. Poincarée Analyse Non Linéeaire 23 (2006), 829–837.Google Scholar
31.Lions, P. L., Symétrie et compacité dans les espaces de Sobolev, J. Funct. Analysis 49 (1982), 315–334.CrossRefGoogle Scholar
32.Rabinowitz, P. H., Minimax methods in critical point theory with applications to differential equations, CBMS Regional Conference Series in Mathematics, Volume 65 (American Mathematical Society, Providence, RI, 1986).CrossRefGoogle Scholar
33.Rabinowitz, P. H., On a class of nonlinear Schródinger equations, Z. Angew. Math. Phys. 43 (1992), 270–291.CrossRefGoogle Scholar
34.Secchi, S., Ground state solutions for nonlinear fractional Schröinger equations in ℝN, J. Math. Phys. 54 (2013), 031501.CrossRefGoogle Scholar
35.Serra, J. and Ros-Oton, X., Fractional Laplacian: Pohozaev identity and nonexistence results, C. R. Acad. Sci. Paris Séer. I 350 (2012), 505–508.Google Scholar
36.Szulkin, A. and Weth, T., Ground state solutions for some indefinite variational problems, J. Funct. Analysis 257 (2009), 3802–3822.CrossRefGoogle Scholar
37.Tan, J. G., The Brezis-Nirenberg type problem involving the square root of the Laplacian, Calc. Var. 42 (2011), 21–41.CrossRefGoogle Scholar