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On the $L^2$-critical nonlinear Schrödinger Equation with a nonlinear damping
Radial symmetry of ground states for a regional fractional Nonlinear Schrödinger Equation
1. | Departamento de Ingeneria Matematica F.C.F.M., Universidad de Chile, Casilla 170 Correro 3, Santiago |
2. | Departamento de Ingeniería Matemática and Centro de Modelamiento Matemático UMR2071 CNRS-UChile, Universidad de Chile, Casilla 170 Correo 3, Santiago, Chile |
References:
[1] |
F. Almgren and E. Lieb, Symmetric decreasing rearrangement is sometimes continuous,, \emph{J. Amer. Math. Soc.}, 2 (1989), 683.
doi: 10.2307/1990893. |
[2] |
W. Beckner, Sobolev Inequalities, the Poisson Semigroup and analysis on the sphere $S^n$,, \emph{Proc. Natl. Acad. Sci.}, 89 (1992), 4816.
doi: 10.1073/pnas.89.11.4816. |
[3] |
H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. I. Existence of a ground state,, \emph{Arch. Rational Mech. Anal.}, 82 (1983), 313.
doi: 10.1007/BF00250555. |
[4] |
R. Blumenthal and R. Getoor, Some theorems on stable processes,, \emph{Trans. Am. Math. Soc.}, 95 (1960), 263.
|
[5] |
K. Bogdan, K. Burdzy and Z. Q. Chen, Censored stable processes,, \emph{Probab. Theory Relat. Fields}, 127 (2003), 89.
doi: 10.1007/s00440-003-0275-1. |
[6] |
M. Cheng, ound state for the fractional Schrödinger equation with unbounded potential,, \emph{J. Math. Phys.}, 53 (2012).
doi: 10.1063/1.3701574. |
[7] |
S. Dipierro, G. Palatucci and E. Valdinoci, Existence and symmetry results for a Schroinger type problem involving the fractional Laplacian,, \emph{Le Matematiche}, LXVIII (2013), 201.
|
[8] |
P. Felmer, A. Quaas and J. Tan, Positive solutions of nonlinear Schrödinguer equation with the fractional laplacian,, Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 142 (2012), 1237.
doi: 10.1017/S0308210511000746. |
[9] |
P. Felmer and C. Torres, Non-linear Schrödinger equation with non-local regional diffusion,, Preprint., (). Google Scholar |
[10] |
Q. Y. Guan, Integration by parts formula for regional fractional Laplacian,, \emph{Commun. Math. Phys.}, 266 (2006), 289.
doi: 10.1007/s00220-006-0054-9. |
[11] |
H. Ishii and G. Nakamura, A class of integral equations and approximation of p-Laplace equations,, \emph{Calc. Var.}, 37 (2010), 485.
doi: 10.1007/s00526-009-0274-x. |
[12] |
L. Jeanjean, On the existence of bounded Palais-Smale sequence and application to a Ladesman.Lazer-type problem set on $\mathbbR^n$,, \emph{Proc. Roy. Soc. Edinburgh Sect. A}, 129 (1999), 787.
doi: 10.1017/S0308210500013147. |
[13] |
B. Kawohl, Rearrangements and Convexity of Level Sets in PDE,, Lect. Notes Math., 1150 (1985).
|
[14] |
S. Kesavan, Symmetrization and Applications,, World Scientific, (2006). Google Scholar |
[15] |
E. Lieb and M. Loss, Analysis,, Grad. Stud. Math., 14 (2001).
|
[16] |
J. Mawhin and M. Willen, Critical Point Theory and Hamiltonian Systems,, Applied Mathematical Sciences, 74 (1989).
doi: 10.1007/978-1-4757-2061-7. |
[17] |
E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces,, \emph{Bull. Sci. Math.}, 136 (2012), 521.
doi: 10.1016/j.bulsci.2011.12.004. |
[18] |
Y. Park, Fractional Polya-Zsego inequality,, \emph{J. Chungcheong Math. Soc.}, 24 (2011), 267. Google Scholar |
[19] |
P. Rabinowitz, On a class of nonlinear Schrödinguer equations,, \emph{ZAMP}, 43 (1992), 270.
doi: 10.1007/BF00946631. |
[20] |
S. Secchi, Ground state solutions for nonlinear fractional Schroinger equations in $\mathbbR^n$,, \emph{J. Math. Phys.}, 54 (2013).
doi: 10.1063/1.4793990. |
[21] |
S. Secchi, On fractional Schrödinger equation in $\mathbbR^n$ without the Ambrosetti-Rabinowitz condition,, to appear in \emph{Topological Methods in Nonlinear Analysis}., (). Google Scholar |
[22] |
B. Simon, Convexity: An Analytic Viewpoint,, Cambridge Tracts in Math. \textbf{187}, 187 (2011).
doi: 10.1017/CBO9780511910135. |
[23] |
J. Van Schaftingen, Symmetrization and minimax principle,, \emph{Comm. Contemporary Math.}, 7 (2005), 463.
doi: 10.1142/S0219199705001817. |
show all references
References:
[1] |
F. Almgren and E. Lieb, Symmetric decreasing rearrangement is sometimes continuous,, \emph{J. Amer. Math. Soc.}, 2 (1989), 683.
doi: 10.2307/1990893. |
[2] |
W. Beckner, Sobolev Inequalities, the Poisson Semigroup and analysis on the sphere $S^n$,, \emph{Proc. Natl. Acad. Sci.}, 89 (1992), 4816.
doi: 10.1073/pnas.89.11.4816. |
[3] |
H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. I. Existence of a ground state,, \emph{Arch. Rational Mech. Anal.}, 82 (1983), 313.
doi: 10.1007/BF00250555. |
[4] |
R. Blumenthal and R. Getoor, Some theorems on stable processes,, \emph{Trans. Am. Math. Soc.}, 95 (1960), 263.
|
[5] |
K. Bogdan, K. Burdzy and Z. Q. Chen, Censored stable processes,, \emph{Probab. Theory Relat. Fields}, 127 (2003), 89.
doi: 10.1007/s00440-003-0275-1. |
[6] |
M. Cheng, ound state for the fractional Schrödinger equation with unbounded potential,, \emph{J. Math. Phys.}, 53 (2012).
doi: 10.1063/1.3701574. |
[7] |
S. Dipierro, G. Palatucci and E. Valdinoci, Existence and symmetry results for a Schroinger type problem involving the fractional Laplacian,, \emph{Le Matematiche}, LXVIII (2013), 201.
|
[8] |
P. Felmer, A. Quaas and J. Tan, Positive solutions of nonlinear Schrödinguer equation with the fractional laplacian,, Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 142 (2012), 1237.
doi: 10.1017/S0308210511000746. |
[9] |
P. Felmer and C. Torres, Non-linear Schrödinger equation with non-local regional diffusion,, Preprint., (). Google Scholar |
[10] |
Q. Y. Guan, Integration by parts formula for regional fractional Laplacian,, \emph{Commun. Math. Phys.}, 266 (2006), 289.
doi: 10.1007/s00220-006-0054-9. |
[11] |
H. Ishii and G. Nakamura, A class of integral equations and approximation of p-Laplace equations,, \emph{Calc. Var.}, 37 (2010), 485.
doi: 10.1007/s00526-009-0274-x. |
[12] |
L. Jeanjean, On the existence of bounded Palais-Smale sequence and application to a Ladesman.Lazer-type problem set on $\mathbbR^n$,, \emph{Proc. Roy. Soc. Edinburgh Sect. A}, 129 (1999), 787.
doi: 10.1017/S0308210500013147. |
[13] |
B. Kawohl, Rearrangements and Convexity of Level Sets in PDE,, Lect. Notes Math., 1150 (1985).
|
[14] |
S. Kesavan, Symmetrization and Applications,, World Scientific, (2006). Google Scholar |
[15] |
E. Lieb and M. Loss, Analysis,, Grad. Stud. Math., 14 (2001).
|
[16] |
J. Mawhin and M. Willen, Critical Point Theory and Hamiltonian Systems,, Applied Mathematical Sciences, 74 (1989).
doi: 10.1007/978-1-4757-2061-7. |
[17] |
E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces,, \emph{Bull. Sci. Math.}, 136 (2012), 521.
doi: 10.1016/j.bulsci.2011.12.004. |
[18] |
Y. Park, Fractional Polya-Zsego inequality,, \emph{J. Chungcheong Math. Soc.}, 24 (2011), 267. Google Scholar |
[19] |
P. Rabinowitz, On a class of nonlinear Schrödinguer equations,, \emph{ZAMP}, 43 (1992), 270.
doi: 10.1007/BF00946631. |
[20] |
S. Secchi, Ground state solutions for nonlinear fractional Schroinger equations in $\mathbbR^n$,, \emph{J. Math. Phys.}, 54 (2013).
doi: 10.1063/1.4793990. |
[21] |
S. Secchi, On fractional Schrödinger equation in $\mathbbR^n$ without the Ambrosetti-Rabinowitz condition,, to appear in \emph{Topological Methods in Nonlinear Analysis}., (). Google Scholar |
[22] |
B. Simon, Convexity: An Analytic Viewpoint,, Cambridge Tracts in Math. \textbf{187}, 187 (2011).
doi: 10.1017/CBO9780511910135. |
[23] |
J. Van Schaftingen, Symmetrization and minimax principle,, \emph{Comm. Contemporary Math.}, 7 (2005), 463.
doi: 10.1142/S0219199705001817. |
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