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$ L^p $-$ L^q $ estimates for the damped wave equation and the critical exponent for the nonlinear problem with slowly decaying data
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 |
$p \& q$ |
$ \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*} $ |
$ \varepsilon >0 $ |
$ s\in (0, 1) $ |
$ 1< p<q<\frac{N}{s} $ |
$ (-\Delta)^{s}_{t} $ |
$ t\in \{p,q\} $ |
$ t $ |
$ V: \mathbb{R} ^{N}\rightarrow \mathbb{R} $ |
$ f: \mathbb{R} \rightarrow \mathbb{R} $ |
$ \mathcal{C} ^{1} $ |
$ \varepsilon $ |
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[7] |
A. Ambrosetti and P. H. Rabinowitz,
Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349-381.
|
[8] |
V. Ambrosio,
Multiple solutions for a fractional $p$-Laplacian equation with sign-changing potential, Electron. J. Diff. Equ., 2016 (2016), 1-12.
|
[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. |
[10] |
V. Ambrosio, Concentrating solutions for a class of nonlinear fractional Schrödinger equations in $ \mathbb{R} ^N$, Rev. Mat. Iberoam., arXiv: 1612.02388. |
[11] |
V. Ambrosio, Fractional $p \& q$ Laplacian problems in $ \mathbb{R} ^N$ with critical growth, Preprint, arXiv: 1801.10449. |
[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.
|
[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. |
[14] |
V. Ambrosio and T. Isernia,
Concentration phenomena for a fractional Schrödinger-Kirchhoff type equation, Math. Methods Appl. Sci., 41 (2018), 615-645.
|
[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. |
[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.
|
[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. |
[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. |
[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. |
[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.
|
[21] |
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. |
[22] |
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. |
[23] |
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. |
[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. |
[25] |
I. Ekeland,
On the variational principle, J. Math. Anal. Appl., 47 (1974), 324-353.
doi: 10.1016/0022-247X(74)90025-0. |
[26] |
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. |
[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. |
[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. |
[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. |
[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. |
[31] |
G. Franzina and G. Palatucci,
Fractional $p$-eigenvalues, Riv. Math. Univ. Parma (N.S.), 5 (2014), 373-386.
|
[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.
|
[33] |
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. |
[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. |
[35] |
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. |
[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. |
[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. |
[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. |
[39] |
E. Lindgren and P. Lindqvist,
Fractional eigenvalues, Calc. Var., 49 (2014), 795-826.
doi: 10.1007/s00526-013-0600-1. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[48] |
P. H. Rabinowitz,
On a class of nonlinear Schrödinger equations, Z. Angew Math. Phys., 43 (1992), 270-291.
doi: 10.1007/BF00946631. |
[49] |
M. Schechter and W. Zou,
Superlinear problems, Pacific J. Math., 214 (2004), 145-160.
doi: 10.2140/pjm.2004.214.145. |
[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. |
[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). |
[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. |
[53] |
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. |
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[7] |
A. Ambrosetti and P. H. Rabinowitz,
Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349-381.
|
[8] |
V. Ambrosio,
Multiple solutions for a fractional $p$-Laplacian equation with sign-changing potential, Electron. J. Diff. Equ., 2016 (2016), 1-12.
|
[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. |
[10] |
V. Ambrosio, Concentrating solutions for a class of nonlinear fractional Schrödinger equations in $ \mathbb{R} ^N$, Rev. Mat. Iberoam., arXiv: 1612.02388. |
[11] |
V. Ambrosio, Fractional $p \& q$ Laplacian problems in $ \mathbb{R} ^N$ with critical growth, Preprint, arXiv: 1801.10449. |
[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.
|
[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. |
[14] |
V. Ambrosio and T. Isernia,
Concentration phenomena for a fractional Schrödinger-Kirchhoff type equation, Math. Methods Appl. Sci., 41 (2018), 615-645.
|
[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. |
[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.
|
[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. |
[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. |
[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. |
[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.
|
[21] |
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. |
[22] |
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. |
[23] |
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. |
[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. |
[25] |
I. Ekeland,
On the variational principle, J. Math. Anal. Appl., 47 (1974), 324-353.
doi: 10.1016/0022-247X(74)90025-0. |
[26] |
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. |
[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. |
[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. |
[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. |
[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. |
[31] |
G. Franzina and G. Palatucci,
Fractional $p$-eigenvalues, Riv. Math. Univ. Parma (N.S.), 5 (2014), 373-386.
|
[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.
|
[33] |
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. |
[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. |
[35] |
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. |
[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. |
[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. |
[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. |
[39] |
E. Lindgren and P. Lindqvist,
Fractional eigenvalues, Calc. Var., 49 (2014), 795-826.
doi: 10.1007/s00526-013-0600-1. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[48] |
P. H. Rabinowitz,
On a class of nonlinear Schrödinger equations, Z. Angew Math. Phys., 43 (1992), 270-291.
doi: 10.1007/BF00946631. |
[49] |
M. Schechter and W. Zou,
Superlinear problems, Pacific J. Math., 214 (2004), 145-160.
doi: 10.2140/pjm.2004.214.145. |
[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. |
[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). |
[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. |
[53] |
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. |
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