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November  2014, 13(6): 2395-2406. doi: 10.3934/cpaa.2014.13.2395

Radial symmetry of ground states for a regional fractional Nonlinear Schrödinger Equation

Patricio Felmer 1, and  César Torres 2,

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

Received  November 2013 Revised  March 2014 Published  July 2014

The aim of this paper is to study radial symmetry properties for ground state solutions of elliptic equations involving a regional fractional Laplacian, namely \begin{eqnarray} (-\Delta)_{\rho}^{\alpha}u + u = f(u) \quad \mbox{in} \ \mathbb{R}^{n}, \ \ \mbox{for} \ \ \alpha\in (0,1). \end{eqnarray} In [9], the authors proved that problem (1) has a ground state solution. In this work we prove that the ground state level is achieved by a radially symmetry solution. The proof is carried out by using variational methods jointly with rearrangement arguments.
Keywords: Radial symmetry, nonlinear Schrödinger equation, fractional Laplacian, regional operator..
Mathematics Subject Classification: Primary: 35J60, 35R11; Secondary: 35J20, 35B0.
Citation: Patricio Felmer, César Torres. Radial symmetry of ground states for a regional fractional Nonlinear Schrödinger Equation. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2395-2406. doi: 10.3934/cpaa.2014.13.2395
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

R. Blumenthal and R. Getoor, Some theorems on stable processes,, \emph{Trans. Am. Math. Soc.}, 95 (1960), 263.   Google Scholar

[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.  Google Scholar

[6]

M. Cheng, ound state for the fractional Schrödinger equation with unbounded potential,, \emph{J. Math. Phys.}, 53 (2012).  doi: 10.1063/1.3701574.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

B. Kawohl, Rearrangements and Convexity of Level Sets in PDE,, Lect. Notes Math., 1150 (1985).   Google Scholar

[14]

S. Kesavan, Symmetrization and Applications,, World Scientific, (2006).   Google Scholar

[15]

E. Lieb and M. Loss, Analysis,, Grad. Stud. Math., 14 (2001).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

J. Van Schaftingen, Symmetrization and minimax principle,, \emph{Comm. Contemporary Math.}, 7 (2005), 463.  doi: 10.1142/S0219199705001817.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

R. Blumenthal and R. Getoor, Some theorems on stable processes,, \emph{Trans. Am. Math. Soc.}, 95 (1960), 263.   Google Scholar

[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.  Google Scholar

[6]

M. Cheng, ound state for the fractional Schrödinger equation with unbounded potential,, \emph{J. Math. Phys.}, 53 (2012).  doi: 10.1063/1.3701574.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

B. Kawohl, Rearrangements and Convexity of Level Sets in PDE,, Lect. Notes Math., 1150 (1985).   Google Scholar

[14]

S. Kesavan, Symmetrization and Applications,, World Scientific, (2006).   Google Scholar

[15]

E. Lieb and M. Loss, Analysis,, Grad. Stud. Math., 14 (2001).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

J. Van Schaftingen, Symmetrization and minimax principle,, \emph{Comm. Contemporary Math.}, 7 (2005), 463.  doi: 10.1142/S0219199705001817.  Google Scholar

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