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Ground state solutions for fractional scalar field equations under a general critical nonlinearity
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 |
$ (- \Delta)^{\alpha}u = g(u) \ \ \mbox{in} \ \ \mathbb{R}^{N}, \ \ u \in H^{\alpha}(\mathbb R^N) $ |
$ (-\Delta)^{\alpha} $ |
$ \alpha\in (0,1) $ |
$ N\geq2 $ |
$ N = 1 $ |
$ \alpha = 1/2 $ |
$ g $ |
$ N\geq2 $ |
$ N = 1 $ |
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. |
[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. |
[3] |
V. Ambrosio,
Zero mass case for a fractional Beresticky-Lions type results, Adv. Nonlinear Anal., 7 (2018), 365-374.
|
[4] |
A unified approach to symmetrization, Partial Differential Equations of Elliptic Type, Symposia Mathematica, Vol. XXXV, Cambridge Univ. Press, 1994, pp. 47–91. |
[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. |
[6] |
H. Berestycki and P. L. Lions,
Nonlinear Scalar Field, Arch. Rational Mech. Anal., 82 (1983), 313-345.
doi: 10.1007/BF00250555. |
[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.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[15] |
I. Ekeland,
On the variational principle, J. Math. Anal. Appl., 47 (1974), 324-355.
doi: 10.1016/0022-247X(74)90025-0. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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.
|
[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. |
[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. |
[24] |
T. Ozawa,
On critical cases of Sobolev's inequalities, J. Funct. Anal., 127 (1995), 259-269.
doi: 10.1006/jfan.1995.1012. |
[25] |
M. Willem, Minimax Theorems, Birkhäuser, Boston
doi: 10.1007/978-1-4612-4146-1. |
[26] |
J. Zhang and W. Zou,
A Berestycki-Lions theorem revisited, Comm. Contemp. Math, 16 (14), 1250033-1.
doi: 10.1142/S0219199712500332. |
[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. |
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. |
[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. |
[3] |
V. Ambrosio,
Zero mass case for a fractional Beresticky-Lions type results, Adv. Nonlinear Anal., 7 (2018), 365-374.
|
[4] |
A unified approach to symmetrization, Partial Differential Equations of Elliptic Type, Symposia Mathematica, Vol. XXXV, Cambridge Univ. Press, 1994, pp. 47–91. |
[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. |
[6] |
H. Berestycki and P. L. Lions,
Nonlinear Scalar Field, Arch. Rational Mech. Anal., 82 (1983), 313-345.
doi: 10.1007/BF00250555. |
[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.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[15] |
I. Ekeland,
On the variational principle, J. Math. Anal. Appl., 47 (1974), 324-355.
doi: 10.1016/0022-247X(74)90025-0. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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.
|
[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. |
[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. |
[24] |
T. Ozawa,
On critical cases of Sobolev's inequalities, J. Funct. Anal., 127 (1995), 259-269.
doi: 10.1006/jfan.1995.1012. |
[25] |
M. Willem, Minimax Theorems, Birkhäuser, Boston
doi: 10.1007/978-1-4612-4146-1. |
[26] |
J. Zhang and W. Zou,
A Berestycki-Lions theorem revisited, Comm. Contemp. Math, 16 (14), 1250033-1.
doi: 10.1142/S0219199712500332. |
[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. |
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