Show Summary Details
More options …

IMPACT FACTOR 2017: 1.029
5-year IMPACT FACTOR: 1.147

CiteScore 2017: 1.29

SCImago Journal Rank (SJR) 2017: 1.588
Source Normalized Impact per Paper (SNIP) 2017: 0.971

Mathematical Citation Quotient (MCQ) 2017: 1.03

Online
ISSN
2169-0375
See all formats and pricing
More options …
Volume 19, Issue 1

# Sign-Changing Solutions for a Class of Zero Mass Nonlocal Schrödinger Equations

Vincenzo Ambrosio
/ Giovany M. Figueiredo
/ Teresa Isernia
/ Giovanni Molica Bisci
• Corresponding author
• Dipartimento P.A.U., Università degli Studi Mediterranea di Reggio Calabria, Salita Melissari - Feo di Vito, 89100 Reggio Calabria, Italy
• Email
• Other articles by this author:
Published Online: 2018-07-07 | DOI: https://doi.org/10.1515/ans-2018-2023

## Abstract

We consider the following class of fractional Schrödinger equations:

where $\alpha \in \left(0,1\right)$, $N>2\alpha$, ${\left(-\mathrm{\Delta }\right)}^{\alpha }$ is the fractional Laplacian, V and K are positive continuous functions which vanish at infinity, and f is a continuous function. By using a minimization argument and a quantitative deformation lemma, we obtain the existence of a sign-changing solution. Furthermore, when f is odd, we prove that the above problem admits infinitely many nontrivial solutions. Our result extends to the fractional framework some well-known theorems proved for elliptic equations in the classical setting. With respect to these cases studied in the literature, the nonlocal one considered here presents some additional difficulties, such as the lack of decompositions involving positive and negative parts, and the non-differentiability of the Nehari Manifold, so that a careful analysis of the fractional spaces involved is necessary.

MSC 2010: 35A15; 35J60; 35R11; 45G05

## References

• [1]

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 (2016), no. 3, Article ID 47. Google Scholar

• [2]

C. O. Alves and M. A. S. Souto, Existence of solutions for a class of nonlinear Schrödinger equations with potential vanishing at infinity, J. Differential Equations 254 (2013), no. 4, 1977–1991.

• [3]

C. O. Alves and M. A. S. Souto, Existence of least energy nodal solution for a Schrödinger–Poisson system in bounded domains, Z. Angew. Math. Phys. 65 (2014), no. 6, 1153–1166.

• [4]

A. Ambrosetti, V. Felli and A. Malchiodi, Ground states of nonlinear Schrödinger equations with potentials vanishing at infinity, J. Eur. Math. Soc. (JEMS) 7 (2005), no. 1, 117–144. Google Scholar

• [5]

A. Ambrosetti and Z.-Q. Wang, Nonlinear Schrödinger equations with vanishing and decaying potentials, Differential Integral Equations 18 (2005), no. 12, 1321–1332. Google Scholar

• [6]

V. Ambrosio, Multiplicity and concentration results for a fractional Choquard equation via penalization method, Potential Anal. (2017), 10.1007/s11118-017-9673-3. Google Scholar

• [7]

V. Ambrosio, Multiplicity of positive solutions for a class of fractional Schrödinger equations via penalization method, Ann. Mat. Pura Appl. (4) 196 (2017), no. 6, 2043–2062.

• [8]

V. Ambrosio, Mountain pass solutions for the fractional Berestycki–Lions problem, Adv. Differential Equations 23 (2018), no. 5–6, 455–488. Google Scholar

• [9]

V. Ambrosio and G. M. Figueiredo, Ground state solutions for a fractional Schrödinger equation with critical growth, Asymptot. Anal. 105 (2017), no. 3–4, 159–191.

• [10]

V. Ambrosio and H. Hajaiej, Multiple solutions for a class of nonhomogeneous fractional Schrödinger equations in ${ℝ}^{ℕ}$, J. Dynam. Differential Equations (2017), 10.1007/s10884-017-9590-6. Google Scholar

• [11]

V. Ambrosio and T. Isernia, A multiplicity result for a fractional Kirchhoff equation in ${ℝ}^{ℕ}$ with a general nonlinearity, Commun. Contemp. Math. (2016), 10.1142/S0219199717500547. Google Scholar

• [12]

V. Ambrosio and T. Isernia, Sign-changing solutions for a class of Schrödinger equations with vanishing potentials, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 29 (2018), no. 1, 127–152.

• [13]

G. Autuori and P. Pucci, Elliptic problems involving the fractional Laplacian in ${ℝ}^{ℕ}$, J. Differential Equations 255 (2013), no. 8, 2340–2362. Google Scholar

• [14]

S. Barile and G. M. Figueiredo, Existence of least energy positive, negative and nodal solutions for a class of $p&q$-problems with potentials vanishing at infinity, J. Math. Anal. Appl. 427 (2015), no. 2, 1205–1233. Google Scholar

• [15]

T. Bartsch, Z. Liu and T. Weth, Sign changing solutions of superlinear Schrödinger equations, Comm. Partial Differential Equations 29 (2004), no. 1–2, 25–42. Google Scholar

• [16]

T. Bartsch, A. Pankov and Z.-Q. Wang, Nonlinear Schrödinger equations with steep potential well, Commun. Contemp. Math. 3 (2001), no. 4, 549–569.

• [17]

T. Bartsch and T. Weth, Three nodal solutions of singularly perturbed elliptic equations on domains without topology, Ann. Inst. H. Poincaré Anal. Non Linéaire 22 (2005), no. 3, 259–281.

• [18]

T. Bartsch, T. Weth and M. Willem, Partial symmetry of least energy nodal solutions to some variational problems, J. Anal. Math. 96 (2005), 1–18.

• [19]

V. Benci, C. R. Grisanti and A. M. Micheletti, Existence of solutions for the nonlinear Schrödinger equation with $V\left(\mathrm{\infty }\right)=0$, Contributions to Nonlinear Analysis, Progr. Nonlinear Differential Equations Appl. 66, Birkhäuser, Basel (2006), 53–65. Google Scholar

• [20]

H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. I. Existence of a ground state, Arch. Ration. Mech. Anal. 82 (1983), no. 4, 313–345.

• [21]

G. Molica Bisci and V. D. Rădulescu, Ground state solutions of scalar field fractional Schrödinger equations, Calc. Var. Partial Differential Equations 54 (2015), no. 3, 2985–3008.

• [22]

D. Bonheure and J. Van Schaftingen, Groundstates for the nonlinear Schrödinger equation with potential vanishing at infinity, Ann. Mat. Pura Appl. (4) 189 (2010), no. 2, 273–301.

• [23]

C. Bucur and E. Valdinoci, Nonlocal Diffusion and Applications, Lect. Notes Unione Mat. Ital. 20, Unione Matematica Italiana, Bologna, 2016. Google Scholar

• [24]

X. Cabré and J. Solà-Morales, Layer solutions in a half-space for boundary reactions, Comm. Pure Appl. Math. 58 (2005), no. 12, 1678–1732.

• [25]

L. Caffarelli, J.-M. Roquejoffre and O. Savin, Nonlocal minimal surfaces, Comm. Pure Appl. Math. 63 (2010), no. 9, 1111–1144. Google Scholar

• [26]

L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations 32 (2007), no. 7–9, 1245–1260.

• [27]

L. A. Caffarelli, S. Salsa and L. Silvestre, Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian, Invent. Math. 171 (2008), no. 2, 425–461.

• [28]

A. Castro, J. Cossio and J. M. Neuberger, A sign-changing solution for a superlinear Dirichlet problem, Rocky Mountain J. Math. 27 (1997), no. 4, 1041–1053.

• [29]

J. Dávila, M. del Pino, S. Dipierro and E. Valdinoci, Concentration phenomena for the nonlocal Schrödinger equation with Dirichlet datum, Anal. PDE 8 (2015), no. 5, 1165–1235.

• [30]

J. Dávila, M. del Pino and J. Wei, Concentrating standing waves for the fractional nonlinear Schrödinger equation, J. Differential Equations 256 (2014), no. 2, 858–892.

• [31]

E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker’s guide to the fractional Sobolev spaces, Bull. Sci. Math. 136 (2012), no. 5, 521–573.

• [32]

S. Dipierro, M. Medina and E. Valdinoci, Fractional elliptic problems with critical growth in the whole of ${ℝ}^{ℕ}$, Appunti. Sc. Norm. Super. Pisa (N. S.) 15, Edizioni della Normale, Pisa, 2017. Google Scholar

• [33]

S. Dipierro, G. Palatucci and E. Valdinoci, Existence and symmetry results for a Schrödinger type problem involving the fractional Laplacian, Mat. (Catania) 68 (2013), no. 1, 201–216. Google Scholar

• [34]

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 (2012), no. 6, 1237–1262.

• [35]

G. M. Figueiredo and J. R. Santos Júnior, Existence of a least energy nodal solution for a Schrödinger–Kirchhoff equation with potential vanishing at infinity, J. Math. Phys. 56 (2015), no. 5, Article ID 051506. Google Scholar

• [36]

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), no. 2, Article ID 12. Google Scholar

• [37]

A. Fiscella, P. Pucci and S. Saldi, Existence of entire solutions for Schrödinger–Hardy systems involving two fractional operators, Nonlinear Anal. 158 (2017), 109–131.

• [38]

R. L. Frank, E. Lenzmann and L. Silvestre, Uniqueness of radial solutions for the fractional Laplacian, Comm. Pure Appl. Math. 69 (2016), no. 9, 1671–1726.

• [39]

F. Gazzola and V. Rădulescu, A nonsmooth critical point theory approach to some nonlinear elliptic equations in ${ℝ}^{n}$, Differential Integral Equations 13 (2000), no. 1–3, 47–60. Google Scholar

• [40]

T. Isernia, Positive solution for nonhomogeneous sublinear fractional equations in ${ℝ}^{N}$, Complex Var. Elliptic Equ. 63 (2018), no. 5,689–714. Google Scholar

• [41]

T. Kuusi, G. Mingione and Y. Sire, Nonlocal equations with measure data, Comm. Math. Phys. 337 (2015), no. 3, 1317–1368.

• [42]

N. Laskin, Fractional quantum mechanics and Lévy path integrals, Phys. Lett. A 268 (2000), no. 4–6, 298–305.

• [43]

N. Laskin, Fractional Schrödinger equation, Phys. Rev. E (3) 66 (2002), no. 5, Article ID 056108. Google Scholar

• [44]

C. Miranda, Un’osservazione su un teorema di Brouwer, Boll. Un. Mat. Ital. (2) 3 (1940), 5–7. Google Scholar

• [45]

G. Molica Bisci, V. D. Radulescu and R. Servadei, Variational Methods for Nonlocal Fractional Problems, Encyclopedia Math. Appl. 162, Cambridge University Press, Cambridge, 2016. Google Scholar

• [46]

P. Pucci and S. Saldi, Multiple solutions for an eigenvalue problem involving non-local elliptic p-Laplacian operators, Geometric Methods in PDE’s, Springer INdAM Ser. 13, Springer, Cham (2015), 159–176. Google Scholar

• [47]

P. Pucci and S. Saldi, Critical stationary Kirchhoff equations in ${ℝ}^{N}$ involving nonlocal operators, Rev. Mat. Iberoam. 32 (2016), no. 1, 1–22. Google Scholar

• [48]

P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Reg. Conf. Ser. Math. 65, American Mathematical Society, Providence, 1986. Google Scholar

• [49]

P. H. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys. 43 (1992), no. 2, 270–291.

• [50]

S. Secchi, Ground state solutions for nonlinear fractional Schrödinger equations in ${ℝ}^{N}$, J. Math. Phys. 54 (2013), no. 3, Article ID 031501. Google Scholar

• [51]

W. A. Strauss, Existence of solitary waves in higher dimensions, Comm. Math. Phys. 55 (1977), no. 2, 149–162.

• [52]

A. Szulkin and T. Weth, The method of Nehari manifold, Handbook of Nonconvex Analysis and Applications, International Press, Somerville (2010), 597–632. Google Scholar

• [53]

M. Willem, Minimax Theorems, Progr. Nonlinear Differential Equations Appl. 24, Birkhäuser, Boston, 1996. Google Scholar

Revised: 2018-02-26

Accepted: 2018-06-03

Published Online: 2018-07-07

Published in Print: 2019-02-01

The manuscript was realized within the auspices of the INdAM–GNAMPA projects 2017 titled “Teoria e modelli per problemi non locali”.

Citation Information: Advanced Nonlinear Studies, Volume 19, Issue 1, Pages 113–132, ISSN (Online) 2169-0375, ISSN (Print) 1536-1365,

Export Citation

© 2018 Walter de Gruyter GmbH, Berlin/Boston.