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September  2019, 18(5): 2199-2215. doi: 10.3934/cpaa.2019099

Ground state solutions for fractional scalar field equations under a general critical nonlinearity

Claudianor O. Alves 1, , Giovany M. Figueiredo 2, and  Gaetano Siciliano 3,,

1. 

Universidade Federal de Campina Grande, 58429-970, Campina Grande - PB - Brazil
2. 

Universidade de Brasilia, Campus Universitário Darcy Ribeiro, 70910-900, Brasília - DF - Brazil
3. 

Universidade de São Paulo, Departamento de Matemática - IME, Rua do Matão 1010, 05508-090, São Paulo - SP - Brazil

* Corresponding author

Received  June 2017 Revised  November 2017 Published  April 2019

Fund Project: The first author is supported was partially supported by CNPq/Brazil Proc. 304036/2013-7; the second author was partially supported by CNPq, Capes and FAPDF, Brazil; the third author was partially supported by CNPq, Capes and FAPESP, Brazil.

In this paper we study existence of ground state solution to the following problem
$ (- \Delta)^{\alpha}u = g(u) \ \ \mbox{in} \ \ \mathbb{R}^{N}, \ \ u \in H^{\alpha}(\mathbb R^N) $
where
$ (-\Delta)^{\alpha} $
 is the fractional Laplacian,
$ \alpha\in (0,1) $
. We treat both cases
$ N\geq2 $
 and
$ N = 1 $
 with
$ \alpha = 1/2 $
. The function
$ g $
 is a general nonlinearity of Berestycki-Lions type which is allowed to have critical growth: polynomial in case
$ N\geq2 $
, exponential if
$ N = 1 $
.
Keywords: Fractional Laplacian, Berestycki-Lions type nonlinearity, critical growth.
Mathematics Subject Classification: Primary: 35J20; Secondary: 35A15, 35B33.
Citation: Claudianor O. Alves, Giovany M. Figueiredo, Gaetano Siciliano. Ground state solutions for fractional scalar field equations under a general critical nonlinearity. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2199-2215. doi: 10.3934/cpaa.2019099
References:
[1]

C. O. Alves, M. Montenegro and M. A. Souto, Existence of a ground state solution for a nonlinear scalar field equation with critical growth, Calc. Var. and PDEs, 43 (2012), 537-554.  doi: 10.1007/s00526-011-0422-y.  Google Scholar

[2]

C. O. Alves, J. M do Ó and O. H. Miyagaki, Concentration phenomena for fractional elliptic equations involving exponential critical growth, Adv. Nonlinear Stud., 16 (2016), 843-861.  doi: 10.1515/ans-2016-0097.  Google Scholar

[3]

V. Ambrosio, Zero mass case for a fractional Beresticky-Lions type results, Adv. Nonlinear Anal., 7 (2018), 365-374.   Google Scholar

[4]

A unified approach to symmetrization, Partial Differential Equations of Elliptic Type, Symposia Mathematica, Vol. XXXV, Cambridge Univ. Press, 1994, pp. 47–91.  Google Scholar

[5]

B. Barrios, E. Colorado, R. Servadei and F. Soria, A critical fractional equation with concave-convex nonlinearities, Ann. Inst. H. Poincaré, Anal. Non Linéaire, 32 (2015), 875-900.  doi: 10.1016/j.anihpc.2014.04.003.  Google Scholar

[6]

H. Berestycki and P. L. Lions, Nonlinear Scalar Field, Arch. Rational Mech. Anal., 82 (1983), 313-345.  doi: 10.1007/BF00250555.  Google Scholar

[7]

H. Berestycki, T. Gallouet and O. Kavian, Equations de Champs scalaires euclidiens non linéaires dans le plan., C. R. Acad. Sci. Paris Ser. I Math., 297 (1984), 307-310.   Google Scholar

[8]

C. Bucur and E. Valdinoci, Nonlocal Diffusion and Applications, Lecture Notes of the Unione Matematica Italiana 20. Cham: Springer; Bologna: UMI (ISBN 978-3-319-28738-6/pbk; 978-3-319-28739-3/ebook). ⅹⅱ, 155 p. (2016). doi: 10.1007/978-3-319-28739-3.  Google Scholar

[9]

D. M. Cao, Nonlinear solutions of semilinear elliptic equations with critical exponent in $\mathbb{R}^2$, Comm. Part. Diff. Equat., 17 (1992), 407-435.  doi: 10.1080/03605309208820848.  Google Scholar

[10]

X. Chang and Z-Q Wang, Ground state of scalar field equations involving a fractional Laplacian with general nonlinearity, Nonlinearity, 26 (2013), 479-494.  doi: 10.1088/0951-7715/26/2/479.  Google Scholar

[11]

E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 512-573.  doi: 10.1016/j.bulsci.2011.12.004.  Google Scholar

[12]

S. Dipierro, M. Medina and E. Valdinoci, Fractional elliptic problems with critical growth in the whole of $\mathbb{R}^n$, Lecture Notes. Appunti. Edizioni della Normale, Scuola Normale di Pisa (2017), arXiv: 1506.01748. doi: 10.1007/978-88-7642-601-8.  Google Scholar

[13]

S. Dipierro, M. Medina, I. Peral and E. Valdinoci, Bifurcation results for a fractional elliptic equation with critical exponent in $\mathbb R^{n}$, Manuscripta Mathematica, (2016).  doi: 10.1007/s00229-016-0878-3.  Google Scholar

[14]

J. M. do Ó, O. H. Miyagaki and M. Squassina, Ground states of nonlocal scalar field equations with Trudinger-Moser critical nonlinearity, Topol. Methods Nonlinear Anal., 48 (2016), 477-492.  doi: 10.12775/tmna.2016.045.  Google Scholar

[15]

I. Ekeland, On the variational principle, J. Math. Anal. Appl., 47 (1974), 324-355.  doi: 10.1016/0022-247X(74)90025-0.  Google Scholar

[16]

R. Frank and E. Lenzmann, Uniqueness of non-linear ground states for fractional Laplacians in $\mathbb R$, Acta Math., 210 (2013), 261-318.  doi: 10.1007/s11511-013-0095-9.  Google Scholar

[17]

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

[18]

A. Iannizzotto and M. Squassina, 1/2-Laplacian problems with exponential nonlinearity, J. Math. Anal. Appl., 414 (2014), 372-385.  doi: 10.1016/j.jmaa.2013.12.059.  Google Scholar

[19]

A. Jeanjean and K. Tanaka, A remark on least energy solutions in $\mathbb R^N$, Proc. Amer. Math. Soc., 131 (2002), 2399-2408.  doi: 10.1090/S0002-9939-02-06821-1.  Google Scholar

[20]

R. Lehrer, L. A. Maia and M. Squassina, Asymptotically linear fractional Schrödinger equations, Complex Variables and Elliptic Equations: An International Journal, 60 (2015), 529-558.  doi: 10.1080/17476933.2014.948434.  Google Scholar

[21]

P. L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. Part Ⅰ., Ann. Inst. Henri Poincaré, Anal. Non Linéaire, 1 (1984), 109-145.   Google Scholar

[22]

P. L. Lions, Symétrie et compacité dans les espaces de Sobolev, J. Funct. Analysis, 49 (1982), 315-334.  doi: 10.1016/0022-1236(82)90072-6.  Google Scholar

[23]

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

[24]

T. Ozawa, On critical cases of Sobolev's inequalities, J. Funct. Anal., 127 (1995), 259-269.  doi: 10.1006/jfan.1995.1012.  Google Scholar

[25]

M. Willem, Minimax Theorems, Birkhäuser, Boston doi: 10.1007/978-1-4612-4146-1.  Google Scholar

[26]

J. Zhang and W. Zou, A Berestycki-Lions theorem revisited, Comm. Contemp. Math, 16 (14), 1250033-1.  doi: 10.1142/S0219199712500332.  Google Scholar

[27]

J. J. Zhang, J. M. do Ó and M. Squassina, Fractional Schrödinger-Poisson systems with a general subcritical or critical nonlinearity, Advanced Nonlinear Studies, 16 (2016), 15-30.  doi: 10.1515/ans-2015-5024.  Google Scholar

show all references

References:
[1]

C. O. Alves, M. Montenegro and M. A. Souto, Existence of a ground state solution for a nonlinear scalar field equation with critical growth, Calc. Var. and PDEs, 43 (2012), 537-554.  doi: 10.1007/s00526-011-0422-y.  Google Scholar

[2]

C. O. Alves, J. M do Ó and O. H. Miyagaki, Concentration phenomena for fractional elliptic equations involving exponential critical growth, Adv. Nonlinear Stud., 16 (2016), 843-861.  doi: 10.1515/ans-2016-0097.  Google Scholar

[3]

V. Ambrosio, Zero mass case for a fractional Beresticky-Lions type results, Adv. Nonlinear Anal., 7 (2018), 365-374.   Google Scholar

[4]

A unified approach to symmetrization, Partial Differential Equations of Elliptic Type, Symposia Mathematica, Vol. XXXV, Cambridge Univ. Press, 1994, pp. 47–91.  Google Scholar

[5]

B. Barrios, E. Colorado, R. Servadei and F. Soria, A critical fractional equation with concave-convex nonlinearities, Ann. Inst. H. Poincaré, Anal. Non Linéaire, 32 (2015), 875-900.  doi: 10.1016/j.anihpc.2014.04.003.  Google Scholar

[6]

H. Berestycki and P. L. Lions, Nonlinear Scalar Field, Arch. Rational Mech. Anal., 82 (1983), 313-345.  doi: 10.1007/BF00250555.  Google Scholar

[7]

H. Berestycki, T. Gallouet and O. Kavian, Equations de Champs scalaires euclidiens non linéaires dans le plan., C. R. Acad. Sci. Paris Ser. I Math., 297 (1984), 307-310.   Google Scholar

[8]

C. Bucur and E. Valdinoci, Nonlocal Diffusion and Applications, Lecture Notes of the Unione Matematica Italiana 20. Cham: Springer; Bologna: UMI (ISBN 978-3-319-28738-6/pbk; 978-3-319-28739-3/ebook). ⅹⅱ, 155 p. (2016). doi: 10.1007/978-3-319-28739-3.  Google Scholar

[9]

D. M. Cao, Nonlinear solutions of semilinear elliptic equations with critical exponent in $\mathbb{R}^2$, Comm. Part. Diff. Equat., 17 (1992), 407-435.  doi: 10.1080/03605309208820848.  Google Scholar

[10]

X. Chang and Z-Q Wang, Ground state of scalar field equations involving a fractional Laplacian with general nonlinearity, Nonlinearity, 26 (2013), 479-494.  doi: 10.1088/0951-7715/26/2/479.  Google Scholar

[11]

E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 512-573.  doi: 10.1016/j.bulsci.2011.12.004.  Google Scholar

[12]

S. Dipierro, M. Medina and E. Valdinoci, Fractional elliptic problems with critical growth in the whole of $\mathbb{R}^n$, Lecture Notes. Appunti. Edizioni della Normale, Scuola Normale di Pisa (2017), arXiv: 1506.01748. doi: 10.1007/978-88-7642-601-8.  Google Scholar

[13]

S. Dipierro, M. Medina, I. Peral and E. Valdinoci, Bifurcation results for a fractional elliptic equation with critical exponent in $\mathbb R^{n}$, Manuscripta Mathematica, (2016).  doi: 10.1007/s00229-016-0878-3.  Google Scholar

[14]

J. M. do Ó, O. H. Miyagaki and M. Squassina, Ground states of nonlocal scalar field equations with Trudinger-Moser critical nonlinearity, Topol. Methods Nonlinear Anal., 48 (2016), 477-492.  doi: 10.12775/tmna.2016.045.  Google Scholar

[15]

I. Ekeland, On the variational principle, J. Math. Anal. Appl., 47 (1974), 324-355.  doi: 10.1016/0022-247X(74)90025-0.  Google Scholar

[16]

R. Frank and E. Lenzmann, Uniqueness of non-linear ground states for fractional Laplacians in $\mathbb R$, Acta Math., 210 (2013), 261-318.  doi: 10.1007/s11511-013-0095-9.  Google Scholar

[17]

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

[18]

A. Iannizzotto and M. Squassina, 1/2-Laplacian problems with exponential nonlinearity, J. Math. Anal. Appl., 414 (2014), 372-385.  doi: 10.1016/j.jmaa.2013.12.059.  Google Scholar

[19]

A. Jeanjean and K. Tanaka, A remark on least energy solutions in $\mathbb R^N$, Proc. Amer. Math. Soc., 131 (2002), 2399-2408.  doi: 10.1090/S0002-9939-02-06821-1.  Google Scholar

[20]

R. Lehrer, L. A. Maia and M. Squassina, Asymptotically linear fractional Schrödinger equations, Complex Variables and Elliptic Equations: An International Journal, 60 (2015), 529-558.  doi: 10.1080/17476933.2014.948434.  Google Scholar

[21]

P. L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. Part Ⅰ., Ann. Inst. Henri Poincaré, Anal. Non Linéaire, 1 (1984), 109-145.   Google Scholar

[22]

P. L. Lions, Symétrie et compacité dans les espaces de Sobolev, J. Funct. Analysis, 49 (1982), 315-334.  doi: 10.1016/0022-1236(82)90072-6.  Google Scholar

[23]

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

[24]

T. Ozawa, On critical cases of Sobolev's inequalities, J. Funct. Anal., 127 (1995), 259-269.  doi: 10.1006/jfan.1995.1012.  Google Scholar

[25]

M. Willem, Minimax Theorems, Birkhäuser, Boston doi: 10.1007/978-1-4612-4146-1.  Google Scholar

[26]

J. Zhang and W. Zou, A Berestycki-Lions theorem revisited, Comm. Contemp. Math, 16 (14), 1250033-1.  doi: 10.1142/S0219199712500332.  Google Scholar

[27]

J. J. Zhang, J. M. do Ó and M. Squassina, Fractional Schrödinger-Poisson systems with a general subcritical or critical nonlinearity, Advanced Nonlinear Studies, 16 (2016), 15-30.  doi: 10.1515/ans-2015-5024.  Google Scholar

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