American Institute of Mathematical Sciences

C. O. Alves, Existence of positive solutions for a problem with lack of compactness involving the $p$-Laplacian, Nonlinear Anal., 51 (2002), 1187-1206. doi: 10.1016/S0362-546X(01)00887-2. Google Scholar

C. O. Alves and V. Ambrosio, A multiplicity result for a nonlinear fractional Schrödinger equation in $ \mathbb{R} ^N$ without the Ambrosetti-Rabinowitz condition, J. Math. Anal. Appl., 466 (2018), 498-522. doi: 10.1016/j.jmaa.2018.06.005. Google Scholar

C. O. Alves and G. M. Figueiredo, Multiplicity and concentration of positive solutions for a class of quasilinear problems, Adv. Nonlinear Stud., 11 (2011), 265-294. doi: 10.1515/ans-2011-0203. Google Scholar

C. O. Alves and O. H. Miyagaki, Existence and concentration of solution for a class of fractional elliptic equation in $\mathbb{R} ^N$ via penalization method, Calc. Var. Partial Differential Equations, 55 (2016), Art. 47, 19 pp. doi: 10.1007/s00526-016-0983-x. Google Scholar

C. O. Alves and M. T. O. Pimenta, On existence and concentration of solutions to a class of quasilinear problems involving the 1-Laplace operator, Calc. Var. Partial Differential Equations, 56 (2017), Art. 143, 24 pp. doi: 10.1007/s00526-017-1236-3. Google Scholar

C. O. Alves and C. L. Torres, Existence and concentration of solution for a non-local regional Schrödinger equation with competing potentials, Glasgow Mathematical Journal, to appear.Google Scholar

A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349-381. Google Scholar

V. Ambrosio, Multiple solutions for a fractional $p$-Laplacian equation with sign-changing potential, Electron. J. Diff. Equ., 2016 (2016), 1-12. Google Scholar

V. Ambrosio, Multiplicity of positive solutions for a class of fractional Schrödinger equations via penalization method, Ann. Mat. Pura Appl., 196 (2017), 2043-2062. doi: 10.1007/s10231-017-0652-5. Google Scholar

V. Ambrosio, Concentrating solutions for a class of nonlinear fractional Schrödinger equations in $ \mathbb{R} ^N$, Rev. Mat. Iberoam., arXiv: 1612.02388.Google Scholar

V. Ambrosio, Fractional $p \& q$ Laplacian problems in $ \mathbb{R} ^N$ with critical growth, Preprint, arXiv: 1801.10449.Google Scholar

V. Ambrosio, A multiplicity result for a fractional $p$-Laplacian problem without growth conditions, Riv. Math. Univ. Parma (N.S.), 9 (2018), 53-71. Google Scholar

V. Ambrosio and H. Hajaiej, Multiple solutions for a class of nonhomogeneous fractional Schrödinger equations in $ \mathbb{R} ^N$, J. Dynam. Differential Equations, 30 (2018), 1119-1143. doi: 10.1007/s10884-017-9590-6. Google Scholar

V. Ambrosio and T. Isernia, Concentration phenomena for a fractional Schrödinger-Kirchhoff type equation, Math. Methods Appl. Sci., 41 (2018), 615-645. Google Scholar

V. Ambrosio and T. Isernia, Sign-changing solutions for a class of Schrödinger equations with vanishing potentials, Rend. Lincei Mat. Appl., 29 (2018), 127-152. doi: 10.4171/RLM/797. Google Scholar

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), 5835-5881. Google Scholar

S. Barile and G. M. Figueiredo, Existence of a least energy nodal solution for a class of $p \& q$-quasilinear elliptic equations, Adv. Nonlinear Stud., 14 (2014), 511-530. doi: 10.1515/ans-2014-0215. Google Scholar

H. Brézis and E. H. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc., 88 (1983), 486-490. doi: 10.2307/2044999. Google Scholar

C. Chen and J. Bao, Existence, nonexistence, and multiplicity of solutions for the fractional $p \& q$-Laplacian equation in $ \mathbb{R} ^N$, Bound. Value Probl., (2016), Paper No. 153, 16 pp. doi: 10.1186/s13661-016-0661-0. Google Scholar

L. Cherfils and V. Il'yasov, On the stationary solutions of generalized reaction diffusion equations with $p \& q$-Laplacian, Commun. Pure Appl. Anal., 1 (2004), 1-14. Google Scholar

A. Di Castro, T. Kuusi and G. Palatucci, Nonlocal Harnack inequalities, J. Funct. Anal., 267 (2014), 1807-1836. doi: 10.1016/j.jfa.2014.05.023. Google Scholar

A. Di Castro, T. Kuusi and G. Palatucci, Local behavior of fractional $p$-minimizers, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016), 1279-1299. doi: 10.1016/j.anihpc.2015.04.003. Google Scholar

E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. math., 136 (2012), 521-573. doi: 10.1016/j.bulsci.2011.12.004. Google Scholar

S. Dipierro, M. Medina and E. Valdinoci, Fractional elliptic problems with critical growth in the whole of $ \mathbb{R} ^n$, Appunti. Scuola Normale Superiore di Pisa (Nuova Serie) [Lecture Notes. Scuola Normale Superiore di Pisa (New Series)], 15. Edizioni della Normale, Pisa, 2017.Google Scholar

I. Ekeland, On the variational principle, J. Math. Anal. Appl., 47 (1974), 324-353. doi: 10.1016/0022-247X(74)90025-0. Google Scholar

P. Felmer, A Quass and J. Tan, Positive solutions of nonlinear Schrödinger equation with the fractional Laplacian, Proc. Royal Soc. Edinburgh A, 142 (2012), 1237-1262. doi: 10.1017/S0308210511000746. Google Scholar

G. M. Figueiredo, Existence of positive solutions for a class of $p \& q$ elliptic problems with critical growth on $ \mathbb{R} ^N$, J. Math. Anal. Appl., 378 (2011), 507-518. doi: 10.1016/j.jmaa.2011.02.017. Google Scholar

G. M. Figueiredo, Existence and multiplicity of solutions for a class of $p \& q$ elliptic problems with critical exponent, Math. Nachr., 286 (2013), 1129-1141. doi: 10.1002/mana.201100237. Google Scholar

G. M. Figueiredo and G. Siciliano, A multiplicity result via Ljusternick-Schnirelmann category and Morse theory for a fractional Schrödinger equation in $ \mathbb{R} ^N$, NoDEA Nonlinear Differential Equations Appl., 23 (2016), Art. 12, 22 pp. doi: 10.1007/s00030-016-0355-4. Google Scholar

A. Fiscella and P. Pucci, Kirchhoff-Hardy fractional problems with lack of compactness, Adv. Nonlinear Stud., 17 (2017), 429-456. doi: 10.1515/ans-2017-6021. Google Scholar

G. Franzina and G. Palatucci, Fractional $p$-eigenvalues, Riv. Math. Univ. Parma (N.S.), 5 (2014), 373-386. Google Scholar

C. He and G. Li, The regularity of weak solutions to nonlinear scalar field elliptic equations containing $p \& q$-Laplacians, Ann. Acad. Sci. Fenn. Math., 33 (2008), 337-371. Google Scholar

A. Iannizzotto, S. Mosconi and M. Squassina, Global Hölder regularity for the fractional $p$-Laplacian, Rev. Mat. Iberoam., 32 (2016), 1353-1392. doi: 10.4171/RMI/921. Google Scholar

L. Jeanjean, On the existence of bounded Palais-Smale sequences and application to a Landesman- Lazer type problem set on $ \mathbb{R} ^N$, Proc. Roy. Soc. Edinburgh Sect.A, 129 (1999), 787-809. doi: 10.1017/S0308210500013147. Google Scholar

T. Kuusi, G. Mingione and Y. Sire, Nonlocal equations with measure data, Comm. Math. Phys., 337 (2015), 1317-1368. doi: 10.1007/s00220-015-2356-2. Google Scholar

N. Laskin, Fractional quantum mechanics and Lèvy path integrals, Phys. Lett. A, 268 (2000), 298-305. doi: 10.1016/S0375-9601(00)00201-2. Google Scholar

G. Li and Z. Guo, Multiple solutions for the $p \& q$-Laplacian problem with critical exponent, Acta Math. Sci. Ser. B Engl. Ed., 29 (2009), 903-918. doi: 10.1016/S0252-9602(09)60089-8. Google Scholar

G. B. Li and X. Liang, The existence of nontrivial solutions to nonlinear elliptic equation of $p$-$q$-Laplacian type on $ \mathbb{R} ^N$, Nonlinear Anal., 71 (2009), 2316-2334. doi: 10.1016/j.na.2009.01.066. Google Scholar

E. Lindgren and P. Lindqvist, Fractional eigenvalues, Calc. Var., 49 (2014), 795-826. doi: 10.1007/s00526-013-0600-1. Google Scholar

Z. Liu and Z. Wang, On the Ambrosetti-Rabinowitz superlinear condition, Advanced Nonlinear Studies, 4 (2004), 563-574. doi: 10.1515/ans-2004-0411. Google Scholar

J. Mawhin and G. Molica Bisci, A Brezis-Nirenberg type result for a nonlocal fractional operator, J. Lond. Math. Soc. (2), 95 (2017), 73–93. doi: 10.1112/jlms.12009. Google Scholar

E. S. Medeiros and K. Perera, Multiplicity of solutions for a quasilinear elliptic problem via the cohomoligical index, Nonlinear Anal., 71 (2009), 3654-3660. doi: 10.1016/j.na.2009.02.013. Google Scholar

C. Mercuri and M. Willem, A global compactness result for the $p$-Laplacian involving critical nonlinearities, Discrete Contin. Dyn. Syst., 28 (2010), 469-493. doi: 10.3934/dcds.2010.28.469. Google Scholar

O. Miyagaki and M. Souto, Superlinear problems without Ambrosetti and Rabinowitz growth condition, J. Differential Equations, 245 (2008), 3628-3638. doi: 10.1016/j.jde.2008.02.035. Google Scholar

G. Molica Bisci, V. Rădulescu and R. Servadei, Variational Methods for Nonlocal Fractional Problems, Cambridge University Press, 162, Cambridge, 2016. doi: 10.1017/CBO9781316282397. Google Scholar

J. Moser, A new proof of De Giorgi's theorem concerning the regularity problem for elliptic differential equations, Comm. Pure Appl. Math., 13 (1960), 457-468. doi: 10.1002/cpa.3160130308. Google Scholar

G. Palatucci and A. Pisante, Improved Sobolev embeddings, profile decomposition, and concentration-compactness for fractional Sobolev spaces, Calc. Var. Partial Differential Equations, 50 (2014), 799-829. doi: 10.1007/s00526-013-0656-y. Google Scholar

P. H. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew Math. Phys., 43 (1992), 270-291. doi: 10.1007/BF00946631. Google Scholar

M. Schechter and W. Zou, Superlinear problems, Pacific J. Math., 214 (2004), 145-160. doi: 10.2140/pjm.2004.214.145. Google Scholar

S. Secchi, Ground state solutions for nonlinear fractional Schrödinger equations in $ \mathbb{R} ^N$, J. Math. Phys., 54 (2013), 031501-17 pages. doi: 10.1063/1.4793990. Google Scholar

A. Szulkin and T. Weth, The method of Nehari manifold, in Handbook of Non-Convex Analysis and Applications, 597–632, Int. Press, Somerville, MA, (2010). Google Scholar

C. E. Torres Ledesma, Existence and symmetry result for fractional p-Laplacian in $ \mathbb{R} ^n$, Commun. Pure Appl. Anal., 16 (2017), 99-113. doi: 10.3934/cpaa.2017004. Google Scholar

M. Willem, Minimax Theorems, Progress in Nonlinear Differential Equations and their Applications, 24. Birkhäuser Boston, Inc., Boston, MA, 1996. doi: 10.1007/978-1-4612-4146-1. Google Scholar