## Rocky Mountain Journal of Mathematics

### Multiplicity results for a fractional Schrodinger equation with potentials

Sofiane Khoutir

#### Abstract

We study a class of nonlinear fractional Schrodinger equations: $(-\Delta )^{s}u+V(x)u=f(x,u) \text { in } \mathbb{R} ^{N},$ where $s\in (0,1)$, $N>2s$, $(-\Delta )^{s}$ stands for the fractional Laplacian. By using a variational approach, we establish the existence of at least one nontrivial solution for the above equation with a general potential $V(x)$ which is allowed to be sign-changing and a sublinear nonlinearity $f(x,u)$. Moreover, by using variational methods and the Moser iteration technique, we prove the existence of infinitely many solutions with $V(x)$ is a nonnegative potential and the nonlinearity $f(x,u)$ is locally sublinear with respect to $u$.

#### Article information

Source
Rocky Mountain J. Math., Volume 49, Number 7 (2019), 2205-2226.

Dates
First available in Project Euclid: 8 December 2019

https://projecteuclid.org/euclid.rmjm/1575774134

Digital Object Identifier
doi:10.1216/RMJ-2019-49-7-2205

Mathematical Reviews number (MathSciNet)
MR4039966

Zentralblatt MATH identifier
07152861

#### Citation

Khoutir, Sofiane. Multiplicity results for a fractional Schrodinger equation with potentials. Rocky Mountain J. Math. 49 (2019), no. 7, 2205--2226. doi:10.1216/RMJ-2019-49-7-2205. https://projecteuclid.org/euclid.rmjm/1575774134

#### References

• N. Laskin, Fractional quantum mechanics and Lévy path integrals, Phys. Lett. A 268 (2000), no. 4-6, 298–305. @article1, MRKEY = MR1755089, AUTHOR = Laskin, Nikolai, TITLE = Fractional quantum mechanics and Lévy path integrals, JOURNAL = Phys. Lett. A, FJOURNAL = Physics Letters. A, VOLUME = 268, YEAR = 2000, NUMBER = 4-6, PAGES = 298–305, ISSN = 0375-9601, MRCLASS = 81S40, MRNUMBER = 1755089, DOI = 10.1016/S0375-9601(00)00201-2, URL =,
• N. Laskin, Fractional Schrödinger equation, Phys. Rev. E 66 (2002), no. 5, art. id. 056108, 7 pp. @article2, MRKEY = MR1948569, AUTHOR = Laskin, Nick, TITLE = Fractional Schrödinger equation, JOURNAL = Phys. Rev. E (3), FJOURNAL = Physical Review E. Statistical, Nonlinear, and Soft Matter Physics, VOLUME = 66, YEAR = 2002, NUMBER = 5, PAGES = 056108, 7, ISSN = 1539-3755, MRCLASS = 81Q05, MRNUMBER = 1948569, DOI = 10.1103/PhysRevE.66.056108, URL =,
• M.F. Furtado, L.A. Maiay and E.S. Medeiros, Positive and nodal solutions for a nonlinear Schrödinger equation with indefinite potential, Adv. Nonlinear Stud. 8 (2008), no. 2, 353–373. @article3, MRKEY = MR2402826, AUTHOR = Furtado, Marcelo F. and Maia, Liliane A. and Medeiros, Everaldo S., TITLE = Positive and nodal solutions for a nonlinear Schrödinger equation with indefinite potential, JOURNAL = Adv. Nonlinear Stud., FJOURNAL = Advanced Nonlinear Studies, VOLUME = 8, YEAR = 2008, NUMBER = 2, PAGES = 353–373, ISSN = 1536-1365, MRCLASS = 35J60 (35B38 35J20 47J30 58E05), MRNUMBER = 2402826, DOI = 10.1515/ans-2008-0207, URL =,
• X.H. Tang, New conditions on nonlinearity for a periodic Schrödinger equation having zero as spectrum, J. Math. Anal. Appl. 413 (2014), no. 1, 392–410. @article4, MRKEY = MR3153592, AUTHOR = Tang, X. H., TITLE = New conditions on nonlinearity for a periodic Schrödinger equation having zero as spectrum, JOURNAL = J. Math. Anal. Appl., FJOURNAL = Journal of Mathematical Analysis and Applications, VOLUME = 413, YEAR = 2014, NUMBER = 1, PAGES = 392–410, ISSN = 0022-247X, MRCLASS = 35J91 (35J20), MRNUMBER = 3153592, DOI = 10.1016/j.jmaa.2013.11.062, URL =,
• X.H. Tang, New super-quadratic conditions on ground state solutions for superlinear Schrödinger equation, Adv. Nonlinear Stud. 14 (2014), no. 2, 349–361. @article5, MRKEY = MR3194360, AUTHOR = Tang, X. H., TITLE = New super-quadratic conditions on ground state solutions for superlinear Schrödinger equation, JOURNAL = Adv. Nonlinear Stud., FJOURNAL = Advanced Nonlinear Studies, VOLUME = 14, YEAR = 2014, NUMBER = 2, PAGES = 361–373, ISSN = 1536-1365, MRCLASS = 35J91 (35J20), MRNUMBER = 3194360, DOI = 10.1515/ans-2014-0208, URL =,
• D. Lv and X. Yang, Nonradial solutions for semilinear Schrödinger equations with sign-changing potential, Electron. J. Qual. Theory Differ. Equ. (2015), art. id. 16, 12 pp. @article7, MRKEY = MR3325919, AUTHOR = Lv, Dingyang and Yang, Xuxin, TITLE = Nonradial solutions for semilinear Schrödinger equations with sign-changing potential, JOURNAL = Electron. J. Qual. Theory Differ. Equ., FJOURNAL = Electronic Journal of Qualitative Theory of Differential Equations, YEAR = 2015, PAGES = No. 16, 12, MRCLASS = 35J91 (35B38 35J20), MRNUMBER = 3325919, DOI = 10.14232/ejqtde.2015.1.16, URL =,
• X.H. Tang and S. Chen, Weak potential conditions for Schrödinger equations with critical nonlinearities, J. Aust. Math. Soc. 100 (2016), no. 2, 272–288. @article9, MRKEY = MR3466542, AUTHOR = Tang, X. H. and Chen, Sitong, TITLE = Weak potential conditions for Schrödinger equations with critical nonlinearities, JOURNAL = J. Aust. Math. Soc., FJOURNAL = Journal of the Australian Mathematical Society, VOLUME = 100, YEAR = 2016, NUMBER = 2, PAGES = 272–288, ISSN = 1446-7887, MRCLASS = 35J91 (35B25 35B33 35J20), MRNUMBER = 3466542, DOI = 10.1017/S1446788715000361, URL =,
• J. Liu, J.F. Liao and C.L. Tang, A positive ground state solution for a class of asymptotically periodic Schrödinger equations, Comput. Math. Appl. 71 (2016), no. 4, 965–976. @article10, MRKEY = MR3461272, AUTHOR = Liu, Jiu and Liao, Jia-Feng and Tang, Chun-Lei, TITLE = A positive ground state solution for a class of asymptotically periodic Schrödinger equations, JOURNAL = Comput. Math. Appl., FJOURNAL = Computers & Mathematics with Applications. An International Journal, VOLUME = 71, YEAR = 2016, NUMBER = 4, PAGES = 965–976, ISSN = 0898-1221, MRCLASS = 35J91 (35B10 35J20), MRNUMBER = 3461272, DOI = 10.1016/j.camwa.2016.01.004, URL =,
• H. Liu, H. Chen and X. Yang, Least energy sign-changing solutions for nonlinear Schrödinger equations with indefinite-sign and vanishing potential, Appl. Math. Lett. 53 (2016), 100–106. @article11, MRKEY = MR3426561, AUTHOR = Liu, Hongliang and Chen, Haibo and Yang, Xiaoxia, TITLE = Least energy sign-changing solutions for nonlinear Schrödinger equations with indefinite-sign and vanishing potential, JOURNAL = Appl. Math. Lett., FJOURNAL = Applied Mathematics Letters. An International Journal of Rapid Publication, VOLUME = 53, YEAR = 2016, PAGES = 100–106, ISSN = 0893-9659, MRCLASS = 35J91 (35J20), MRNUMBER = 3426561, DOI = 10.1016/j.aml.2015.10.010, URL =,
• M.F. Furtado and R. Marchi, Existence of solutions to asymptotically periodic Schrödinger equations, Electron. J. Differential Equations (2017), art. id. 15, 7 pp. @article12, MRKEY = MR3609143, AUTHOR = Furtado, Marcelo F. and Marchi, Reinaldo de, TITLE = Existence of solutions to asymptotically periodic Schrödinger equations, JOURNAL = Electron. J. Differential Equations, FJOURNAL = Electronic Journal of Differential Equations, YEAR = 2017, PAGES = Paper No. 15, 7, MRCLASS = 35J91 (35B10 35B38 35J20), MRNUMBER = 3609143,
• H. Shi and H. Chen, Multiple solutions for fractional Schrödinger equations, Electron. J. Differential Equations (2015), art. id. 25, 11 pp. @article13, MRKEY = MR3335755, AUTHOR = Shi, Hongxia and Chen, Haibo, TITLE = Multiple solutions for fractional Schrödinger equations, JOURNAL = Electron. J. Differential Equations, FJOURNAL = Electronic Journal of Differential Equations, YEAR = 2015, PAGES = No. 25, 11, MRCLASS = 35R11 (35B38), MRNUMBER = 3335755,
• W. Zhang, X. Tang and J. Zhang, Infinitely many radial and non-radial solutions for a fractional Schrödinger equation, Comput. Math. Appl. 71 (2016), no. 3, 737–747. @article14, MRKEY = MR3457376, AUTHOR = Zhang, Wen and Tang, Xianhua and Zhang, Jian, TITLE = Infinitely many radial and non-radial solutions for a fractional Schrödinger equation, JOURNAL = Comput. Math. Appl., FJOURNAL = Computers & Mathematics with Applications. An International Journal, VOLUME = 71, YEAR = 2016, NUMBER = 3, PAGES = 737–747, ISSN = 0898-1221, MRCLASS = 35R11 (35A15), MRNUMBER = 3457376, DOI = 10.1016/j.camwa.2015.12.036, URL =,
• S. Khoutir and H. Chen, Existence of infinitely many high energy solutions for a fractional Schrödinger equation in $\R^N$, Appl. Math. Lett. 61 (2016), 156–162. @article15, MRKEY = MR3518463, AUTHOR = Khoutir, Sofiane and Chen, Haibo, TITLE = Existence of infinitely many high energy solutions for a fractional Schrödinger equation in $\Bbb{R}^N$, JOURNAL = Appl. Math. Lett., FJOURNAL = Applied Mathematics Letters. An International Journal of Rapid Publication, VOLUME = 61, YEAR = 2016, PAGES = 156–162, ISSN = 0893-9659, MRCLASS = 35R11 (26A33 35A15), MRNUMBER = 3518463, DOI = 10.1016/j.aml.2016.06.001, URL =,
• K. Teng, Multiple solutions for a class of fractional Schrödinger equations in $\R^N$, Nonlinear Anal. Real World Appl. 21 (2015), 76–86. @article16, MRKEY = MR3261580, AUTHOR = Teng, Kaimin, TITLE = Multiple solutions for a class of fractional Schrödinger equations in $\Bbb R^N$, JOURNAL = Nonlinear Anal. Real World Appl., FJOURNAL = Nonlinear Analysis. Real World Applications. An International Multidisciplinary Journal, VOLUME = 21, YEAR = 2015, PAGES = 76–86, ISSN = 1468-1218, MRCLASS = 35R11 (35A15), MRNUMBER = 3261580, DOI = 10.1016/j.nonrwa.2014.06.008, URL =,
• B. Ge, Multiple solutions of nonlinear Schrödinger equation with fractional Laplacian, Nonlinear Anal. Real World Appl. 30 (2016), 236–247. @article17, MRKEY = MR3464134, AUTHOR = Ge, Bin, TITLE = Multiple solutions of nonlinear Schrödinger equation with the fractional Laplacian, JOURNAL = Nonlinear Anal. Real World Appl., FJOURNAL = Nonlinear Analysis. Real World Applications. An International Multidisciplinary Journal, VOLUME = 30, YEAR = 2016, PAGES = 236–247, ISSN = 1468-1218, MRCLASS = 35R11 (35A01 35A15 35J91), MRNUMBER = 3464134, DOI = 10.1016/j.nonrwa.2016.01.003, URL =,
• M. Du and L. Tian, Infinitely many solutions of the nonlinear fractional Schrödinger equation, Discrete Contin. Dyn. Syst. Ser. B 21 (2016), no. 10, 3407–3428. @article18, MRKEY = MR3593431, AUTHOR = Du, Miao and Tian, Lixin, TITLE = Infinitely many solutions of the nonlinear fractional Schrödinger equations, JOURNAL = Discrete Contin. Dyn. Syst. Ser. B, FJOURNAL = Discrete and Continuous Dynamical Systems. Series B. A Journal Bridging Mathematics and Sciences, VOLUME = 21, YEAR = 2016, NUMBER = 10, PAGES = 3407–3428, ISSN = 1531-3492, MRCLASS = 35R11 (35A15 35S05), MRNUMBER = 3593431, DOI = 10.3934/dcdsb.2016104, URL =,
• C. Chen, Infinitely many solutions for fractional Schrödinger equations in $\R^N$, Electron. J. Differential Equations (2016), art. id. 88, 15 pp. @article19, MRKEY = MR3489980, AUTHOR = Chen, Caisheng, TITLE = Infinitely many solutions for fractional Schrödinger equations in $\Bbb R^N$, JOURNAL = Electron. J. Differential Equations, FJOURNAL = Electronic Journal of Differential Equations, YEAR = 2016, PAGES = Paper No. 88, 15, MRCLASS = 35R11 (35A15 35Q55 47G20), MRNUMBER = 3489980,
• Z. Wang and H.-S. Zhou, Radial sign-changing solution for fractional Schrödinger equation, Discrete Contin. Dyn. Syst. 36 (2016), no. 1, 499–508. @article20, MRKEY = MR3369233, AUTHOR = Wang, Zhengping and Zhou, Huan-Song, TITLE = Radial sign-changing solution for fractional Schrödinger equation, JOURNAL = Discrete Contin. Dyn. Syst., FJOURNAL = Discrete and Continuous Dynamical Systems. Series A, VOLUME = 36, YEAR = 2016, NUMBER = 1, PAGES = 499–508, ISSN = 1078-0947, MRCLASS = 35R11 (35A15 35B07 35J20 35J61), MRNUMBER = 3369233, MRREVIEWER = César Enrique Torres Ledesma, DOI = 10.3934/dcds.2016.36.499, URL =,
• K. Perera, M. Squassina and Y. Yang, Critical fractional $p$-Laplacian problems with possibly vanishing potentials, J. Math. Anal. Appl. 433 (2016), no. 2, 818–831. @article21, MRKEY = MR3398738, AUTHOR = Perera, Kanishka and Squassina, Marco and Yang, Yang, TITLE = Critical fractional $p$-Laplacian problems with possibly vanishing potentials, JOURNAL = J. Math. Anal. Appl., FJOURNAL = Journal of Mathematical Analysis and Applications, VOLUME = 433, YEAR = 2016, NUMBER = 2, PAGES = 818–831, ISSN = 0022-247X, MRCLASS = 35R11 (35A01 35B33 35J92), MRNUMBER = 3398738, DOI = 10.1016/j.jmaa.2015.08.024, URL =,
• J.M. do Ó, O.H. Miyagaki and M. Squassina, Critical and subcritical fractional problems with vanishing potentials, Commun. Contemp. Math. 18 (2016), no. 6, art. id. 1550063, 20 pp. @article22, MRKEY = MR3561958, AUTHOR = do Ó, João Marcos and Miyagaki, Olímpio H. and Squassina, Marco, TITLE = Critical and subcritical fractional problems with vanishing potentials, JOURNAL = Commun. Contemp. Math., FJOURNAL = Communications in Contemporary Mathematics, VOLUME = 18, YEAR = 2016, NUMBER = 6, PAGES = 1550063, 20, ISSN = 0219-1997, MRCLASS = 35R11 (35P15 35P30), MRNUMBER = 3561958, DOI = 10.1142/S0219199715500637, URL =,
• V. Ambrosio, G. Figueiredo, T. Isernia and G. Molica Bisci, Sign-changing solutions for a class of zero mass nonlocal Schrödinger equations, Adv. Nonlinear Stud. 19 (2019), no. 1, 113–132. @article23, MRKEY = MR3912424, AUTHOR = Ambrosio, Vincenzo and Figueiredo, Giovany M. and Isernia, Teresa and Molica Bisci, Giovanni, TITLE = Sign-changing solutions for a class of zero mass nonlocal Schrödinger equations, JOURNAL = Adv. Nonlinear Stud., FJOURNAL = Advanced Nonlinear Studies, VOLUME = 19, YEAR = 2019, NUMBER = 1, PAGES = 113–132, ISSN = 1536-1365, MRCLASS = 35R11 (35A15 35J60 45K05), MRNUMBER = 3912424, MRREVIEWER = Andrey I. Zahariev, DOI = 10.1515/ans-2018-2023, URL =,
• P. Li and Y. Shang, Infinitely many solutions for fractional Schrödinger equations with perturbation via variational methods, Open Math. 15 (2017), 578–586. @article24, MRKEY = MR3649214, AUTHOR = Li, Peiluan and Shang, Youlin, TITLE = Infinitely many solutions for fractional Schrödinger equations with perturbation via variational methods, JOURNAL = Open Math., FJOURNAL = Open Mathematics, VOLUME = 15, YEAR = 2017, NUMBER = 1, PAGES = 578–586, MRCLASS = 35A15 (26A33 35B20), MRNUMBER = 3649214, DOI = 10.1515/math-2017-0053, URL =,
• 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), no. 3, 465–482. @article25, MRKEY = MR3600204, AUTHOR = Pu, Yang and Liu, Jiu and Tang, Chun-Lei, TITLE = Existence of weak solutions for a class of fractional Schrödinger equations with periodic potential, JOURNAL = Comput. Math. Appl., FJOURNAL = Computers & Mathematics with Applications. An International Journal, VOLUME = 73, YEAR = 2017, NUMBER = 3, PAGES = 465–482, ISSN = 0898-1221, MRCLASS = 35R11 (35D30), MRNUMBER = 3600204, DOI = 10.1016/j.camwa.2016.12.004, URL =,
• Q. Wang, Multiple positive solutions of fractional elliptic equations involving concave and convexe nonlinearities in $\R^N$, Commun. Pure Appl. Anal. 15 (2016), no. 5, 1671–1688. @article26, MRKEY = MR3538875, AUTHOR = Wang, Qingfang, TITLE = Multiple positive solutions of fractional elliptic equations involving concave and convex nonlinearities in $\Bbb R^N$, JOURNAL = Commun. Pure Appl. Anal., FJOURNAL = Communications on Pure and Applied Analysis, VOLUME = 15, YEAR = 2016, NUMBER = 5, PAGES = 1671–1688, ISSN = 1534-0392, MRCLASS = 35R11 (35J20 35J61), MRNUMBER = 3538875, DOI = 10.3934/cpaa.2016008, URL =,
• Z. Gao, X.H. Tang and W. Zhang, Multiplicity and concentration of solutions for fractional Schrödinger equation, Taiwan. J. Math. 21 (2017), no. 1, 187–210. @article27, MRKEY = MR3613980, AUTHOR = Gao, Zu and Tang, Xianhua and Zhang, Wen, TITLE = Multiplicity and concentration of solutions for fractional Schrödinger equations, JOURNAL = Taiwanese J. Math., FJOURNAL = Taiwanese Journal of Mathematics, VOLUME = 21, YEAR = 2017, NUMBER = 1, PAGES = 187–210, ISSN = 1027-5487, MRCLASS = 35R11 (26A33 58E30), MRNUMBER = 3613980, DOI = 10.11650/tjm.0.0.7147, URL =,
• L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations 32 (2007), 1245–1260. @articlecafar, MRKEY = MR2354493, AUTHOR = Caffarelli, Luis and Silvestre, Luis, TITLE = An extension problem related to the fractional Laplacian, JOURNAL = Comm. Partial Differential Equations, FJOURNAL = Communications in Partial Differential Equations, VOLUME = 32, YEAR = 2007, NUMBER = 7-9, PAGES = 1245–1260, ISSN = 0360-5302, MRCLASS = 35J70, MRNUMBER = 2354493, MRREVIEWER = Francesco Petitta, DOI = 10.1080/03605300600987306, URL =,
• V. Ambrosio and T. Isernia, Multiplicity and concentration results for some nonlinear Schrödinger equations with the fractional p-Laplacian, Discrete Contin. Dyn. Syst. 38 (2018), no. 11, 5835–5881. @articleAmb1, MRKEY = MR3917790, AUTHOR = Ambrosio, Vincenzo and Isernia, Teresa, TITLE = Multiplicity and concentration results for some nonlinear Schrödinger equations with the fractional $p$-Laplacian, JOURNAL = Discrete Contin. Dyn. Syst., FJOURNAL = Discrete and Continuous Dynamical Systems. Series A, VOLUME = 38, YEAR = 2018, NUMBER = 11, PAGES = 5835–5881, ISSN = 1078-0947, MRCLASS = 35R11 (47G20 58E05), MRNUMBER = 3917790, MRREVIEWER = Rodica Luca, DOI = 10.3934/dcds.2018254, URL =,
• C.O. Alves and V. Ambrosio, A multiplicity result for a nonlinear fractional Schrödinger in $\R^N$ without the Ambrosetti-Rabinowitz condition, J. Math. Anal. Appl. 466 (2018), 498–522. @articleAmb2, MRKEY = MR3818129, AUTHOR = Alves, Claudianor O. and Ambrosio, Vincenzo, TITLE = A multiplicity result for a nonlinear fractional Schrödinger equation in $\Bbb{R}^N$ without the Ambrosetti-Rabinowitz condition, JOURNAL = J. Math. Anal. Appl., FJOURNAL = Journal of Mathematical Analysis and Applications, VOLUME = 466, YEAR = 2018, NUMBER = 1, PAGES = 498–522, ISSN = 0022-247X, MRCLASS = 35R11 (35A15 35B38), MRNUMBER = 3818129, DOI = 10.1016/j.jmaa.2018.06.005, URL =,
• V. Ambrosio and H. Hajaiej, Multiple solutions for a class of nonhomogeneous fractional Schrödinger equations in $\R^N$, J. Dynam. Differential Equations 30 (2018), no. 3, 1119–1143. @articleAmb3, MRKEY = MR3842142, AUTHOR = Ambrosio, Vincenzo and Hajaiej, Hichem, TITLE = Multiple solutions for a class of nonhomogeneous fractional Schrödinger equations in $\Bbb{R}^N$, JOURNAL = J. Dynam. Differential Equations, FJOURNAL = Journal of Dynamics and Differential Equations, VOLUME = 30, YEAR = 2018, NUMBER = 3, PAGES = 1119–1143, ISSN = 1040-7294, MRCLASS = 35R11 (35A15 35J60 45G05), MRNUMBER = 3842142, DOI = 10.1007/s10884-017-9590-6, URL =,
• 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. @articleAmb4, MRKEY = MR3714753, AUTHOR = Ambrosio, Vincenzo, TITLE = Multiplicity of positive solutions for a class of fractional Schrödinger equations via penalization method, JOURNAL = Ann. Mat. Pura Appl. (4), FJOURNAL = Annali di Matematica Pura ed Applicata. Series IV, VOLUME = 196, YEAR = 2017, NUMBER = 6, PAGES = 2043–2062, ISSN = 0373-3114, MRCLASS = 35R11 (35A15 35J60 45G05), MRNUMBER = 3714753, DOI = 10.1007/s10231-017-0652-5, URL =,
• L. Li and X. Zhong, Infinitely many small solutions for the Kirchhoff equation with local sublinear nonlinearities, J. Math. Anal. Appl. 435 (2016), 955–967. @articleLin, MRKEY = MR3423438, AUTHOR = Li, Lin and Zhong, Xin, TITLE = Infinitely many small solutions for the Kirchhoff equation with local sublinear nonlinearities, JOURNAL = J. Math. Anal. Appl., FJOURNAL = Journal of Mathematical Analysis and Applications, VOLUME = 435, YEAR = 2016, NUMBER = 1, PAGES = 955–967, ISSN = 0022-247X, MRCLASS = 35J60 (35J20), MRNUMBER = 3423438, DOI = 10.1016/j.jmaa.2015.10.075, URL =,
• C. Brändle, E. Colorado, A. de Pablo and U. Sánchez, A concave-convex elliptic problem involving the fractional Laplacian, Proc. Roy. Soc. Edinburgh Sect. A. 143 (2013), 39–71. @articlebrand, MRKEY = MR3023003, AUTHOR = Brändle, C. and Colorado, E. and de Pablo, A. and Sánchez, U., TITLE = A concave-convex elliptic problem involving the fractional Laplacian, JOURNAL = Proc. Roy. Soc. Edinburgh Sect. A, FJOURNAL = Proceedings of the Royal Society of Edinburgh. Section A. Mathematics, VOLUME = 143, YEAR = 2013, NUMBER = 1, PAGES = 39–71, ISSN = 0308-2105, MRCLASS = 35R11 (35B45 35J91), MRNUMBER = 3023003, MRREVIEWER = Xavier Ros-Oton, DOI = 10.1017/S0308210511000175, URL =,
• J. Moser, A new proof of De Giorgi's theorem concerning the regularity problem for elliptic differential equations, Commun. Pure Appl. Math. 13 (1960), 457–468. @articleMoser, MRKEY = MR170091, AUTHOR = Moser, Jürgen, TITLE = A new proof of De Giorgi's theorem concerning the regularity problem for elliptic differential equations, JOURNAL = Comm. Pure Appl. Math., FJOURNAL = Communications on Pure and Applied Mathematics, VOLUME = 13, YEAR = 1960, PAGES = 457–468, ISSN = 0010-3640, MRCLASS = 35.42, MRNUMBER = 170091, MRREVIEWER = G. L. Tautz, DOI = 10.1002/cpa.3160130308, URL =,
• P.H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations, CBMS Reg. Conf. Ser. in Math. 65, Amer. Math. Soc. (1986). @book28, MRKEY = MR845785, AUTHOR = Rabinowitz, Paul H., TITLE = Minimax methods in critical point theory with applications to differential equations, SERIES = CBMS Regional Conference Series in Mathematics, VOLUME = 65, PUBLISHER = Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, YEAR = 1986, PAGES = viii+100, ISBN = 0-8218-0715-3, MRCLASS = 58E05 (34C35 47H15 47H17 58F05), MRNUMBER = 845785, MRREVIEWER = Michele Matzeu, DOI = 10.1090/cbms/065, URL =,
• E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhikers guide to the fractional Sobolev spaces, Bull. Sci. Math. 136 (2012), no. 5, 521–573. @article29, MRKEY = MR2944369, AUTHOR = Di Nezza, Eleonora and Palatucci, Giampiero and Valdinoci, Enrico, TITLE = Hitchhiker's guide to the fractional Sobolev spaces, JOURNAL = Bull. Sci. Math., FJOURNAL = Bulletin des Sciences Mathématiques, VOLUME = 136, YEAR = 2012, NUMBER = 5, PAGES = 521–573, ISSN = 0007-4497, MRCLASS = 46E35 (35A23 35S05 35S30), MRNUMBER = 2944369, MRREVIEWER = Lanzhe Liu, DOI = 10.1016/j.bulsci.2011.12.004, URL =,
• R. Kajikiya, A critical point theorem related to the symmetric mountain pass lemma and its applications to elliptic equations, J. Funct. Anal. 225 (2005), 352–370. @article30, MRKEY = MR2152503, AUTHOR = Kajikiya, Ryuji, TITLE = A critical point theorem related to the symmetric mountain pass lemma and its applications to elliptic equations, JOURNAL = J. Funct. Anal., FJOURNAL = Journal of Functional Analysis, VOLUME = 225, YEAR = 2005, NUMBER = 2, PAGES = 352–370, ISSN = 0022-1236, MRCLASS = 58E05 (35B38 35J20 35J60 47J30), MRNUMBER = 2152503, MRREVIEWER = Enrico Serra, DOI = 10.1016/j.jfa.2005.04.005, URL =,