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July  2019, 18(4): 2009-2045. doi: 10.3934/cpaa.2019091

Existence, multiplicity and concentration for a class of fractional $ p \& q $ Laplacian problems in $ \mathbb{R} ^{N} $

1. 

Universidade Federal de Campina Grande, Unidade Academica de Matematica, CEP: 58429-900, Campina Grande-PB, Brazil

2. 

Dipartimento di Ingegneria Industriale e Scienze Matematiche, Università Politecnica delle Marche, Via Brecce Bianche, 12, 60131 Ancona, Italy

* Corresponding author

Received  September 2018 Revised  September 2018 Published  January 2019

In this work we consider the following class of fractional
$p \& q$
Laplacian problems
$ \begin{equation*} (-\Delta)_{p}^{s}u+ (-\Delta)_{q}^{s}u + V( \varepsilon x) (|u|^{p-2}u + |u|^{q-2}u) = f(u) \mbox{ in } \mathbb{R} ^{N}, \end{equation*} $
where
$ \varepsilon >0 $
is a parameter,
$ s\in (0, 1) $
,
$ 1< p<q<\frac{N}{s} $
,
$ (-\Delta)^{s}_{t} $
, with
$ t\in \{p,q\} $
, is the fractional
$ t $
-Laplacian operator,
$ V: \mathbb{R} ^{N}\rightarrow \mathbb{R} $
is a continuous potential and
$ f: \mathbb{R} \rightarrow \mathbb{R} $
is a
$ \mathcal{C} ^{1} $
-function with subcritical growth. Applying minimax theorems and the Ljusternik-Schnirelmann theory, we investigate the existence, multiplicity and concentration of nontrivial solutions provided that
$ \varepsilon $
is sufficiently small.
Citation: Claudianor O. Alves, Vincenzo Ambrosio, Teresa Isernia. Existence, multiplicity and concentration for a class of fractional $ p \& q $ Laplacian problems in $ \mathbb{R} ^{N} $. Communications on Pure & Applied Analysis, 2019, 18 (4) : 2009-2045. doi: 10.3934/cpaa.2019091
References:
[1]

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

[2]

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

[3]

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

[4]

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

[5]

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

[6]

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

[7]

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

[8]

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

[9]

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

[10]

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

[11]

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

[12]

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

[13]

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

[14]

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

[15]

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

[16]

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

[17]

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

[18]

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

[19]

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

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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

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

[22]

A. Di CastroT. 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

[23]

E. Di NezzaG. 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

[24]

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

[25]

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

[26]

P. FelmerA 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

[27]

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

[28]

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

[29]

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

[30]

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

[31]

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

[32]

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

[33]

A. IannizzottoS. 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

[34]

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

[35]

T. KuusiG. 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

[36]

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

[37]

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

[38]

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

[39]

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

[40]

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

[41]

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

[42]

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

[43]

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

[44]

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

[45]

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

[46]

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

[47]

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

[48]

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

[49]

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

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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

[51]

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

[52]

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

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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

show all references

References:
[1]

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

[2]

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

[3]

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

[4]

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

[5]

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

[6]

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

[7]

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

[8]

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

[9]

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

[10]

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

[11]

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

[12]

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

[13]

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

[14]

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

[15]

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

[16]

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

[17]

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

[18]

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

[19]

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

[20]

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

[21]

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

[22]

A. Di CastroT. 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

[23]

E. Di NezzaG. 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

[24]

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

[25]

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

[26]

P. FelmerA 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

[27]

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

[28]

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

[29]

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

[30]

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

[31]

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

[32]

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

[33]

A. IannizzottoS. 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

[34]

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

[35]

T. KuusiG. 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

[36]

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

[37]

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

[38]

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

[39]

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

[40]

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

[41]

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

[42]

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

[43]

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

[44]

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

[45]

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

[46]

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

[47]

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

[48]

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

[49]

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

[50]

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

[51]

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

[52]

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