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

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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

Abstract
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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.
[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., ().
[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).
[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.
[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}., ().
[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., ().
[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).
[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.
[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}., ().
[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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