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 and 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, J. Amer. Math. Soc., 2 (1989), 683-773. doi: 10.2307/1990893.
[2]	W. Beckner, Sobolev Inequalities, the Poisson Semigroup and analysis on the sphere $S^n$, Proc. Natl. Acad. Sci., 89 (1992), 4816-4819. doi: 10.1073/pnas.89.11.4816.
[3]	H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. I. Existence of a ground state, Arch. Rational Mech. Anal., 82 (1983), 313-345. doi: 10.1007/BF00250555.
[4]	R. Blumenthal and R. Getoor, Some theorems on stable processes, Trans. Am. Math. Soc., 95 (1960), 263-273.
[5]	K. Bogdan, K. Burdzy and Z. Q. Chen, Censored stable processes, Probab. Theory Relat. Fields, 127 (2003), 89-152. doi: 10.1007/s00440-003-0275-1.
[6]	M. Cheng, ound state for the fractional Schrödinger equation with unbounded potential, J. Math. Phys., 53 (2012), 043507. 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, Le Matematiche, LXVIII (2013), 201-216.
[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), 1237-1262. 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, Commun. Math. Phys., 266 (2006), 289-329. doi: 10.1007/s00220-006-0054-9.
[11]	H. Ishii and G. Nakamura, A class of integral equations and approximation of p-Laplace equations, Calc. Var., 37 (2010), 485-522. 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 $\mathbbR^n$, Proc. Roy. Soc. Edinburgh Sect. A, 129 (1999), 787-809. doi: 10.1017/S0308210500013147.
[13]	B. Kawohl, Rearrangements and Convexity of Level Sets in PDE, Lect. Notes Math., 1150, Springer-Verlag, Berlin (1985).
[14]	S. Kesavan, Symmetrization and Applications, World Scientific, Hackensack, NJ, 2006.
[15]	E. Lieb and M. Loss, Analysis, Grad. Stud. Math., vol. 14, Amer. Math. Soc., Providence, RI, 2001.
[16]	J. Mawhin and M. Willen, Critical Point Theory and Hamiltonian Systems, Applied Mathematical Sciences, 74, Springer, Berlin, 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, Bull. Sci. Math., 136 (2012), 521-573. doi: 10.1016/j.bulsci.2011.12.004.
[18]	Y. Park, Fractional Polya-Zsego inequality, J. Chungcheong Math. Soc., 24 (2011), 267-271.
[19]	P. Rabinowitz, On a class of nonlinear Schrödinguer equations, ZAMP, 43 (1992), 270-291. doi: 10.1007/BF00946631.
[20]	S. Secchi, Ground state solutions for nonlinear fractional Schroinger equations in $\mathbbR^n$, J. Math. Phys., 54 (2013), 031501. 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. 187, Cambridge University Press, 2011. doi: 10.1017/CBO9780511910135.
[23]	J. Van Schaftingen, Symmetrization and minimax principle, Comm. Contemporary Math., 7 (2005), 463-481. doi: 10.1142/S0219199705001817.

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References:

[1]	F. Almgren and E. Lieb, Symmetric decreasing rearrangement is sometimes continuous, J. Amer. Math. Soc., 2 (1989), 683-773. doi: 10.2307/1990893.
[2]	W. Beckner, Sobolev Inequalities, the Poisson Semigroup and analysis on the sphere $S^n$, Proc. Natl. Acad. Sci., 89 (1992), 4816-4819. doi: 10.1073/pnas.89.11.4816.
[3]	H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. I. Existence of a ground state, Arch. Rational Mech. Anal., 82 (1983), 313-345. doi: 10.1007/BF00250555.
[4]	R. Blumenthal and R. Getoor, Some theorems on stable processes, Trans. Am. Math. Soc., 95 (1960), 263-273.
[5]	K. Bogdan, K. Burdzy and Z. Q. Chen, Censored stable processes, Probab. Theory Relat. Fields, 127 (2003), 89-152. doi: 10.1007/s00440-003-0275-1.
[6]	M. Cheng, ound state for the fractional Schrödinger equation with unbounded potential, J. Math. Phys., 53 (2012), 043507. 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, Le Matematiche, LXVIII (2013), 201-216.
[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), 1237-1262. 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, Commun. Math. Phys., 266 (2006), 289-329. doi: 10.1007/s00220-006-0054-9.
[11]	H. Ishii and G. Nakamura, A class of integral equations and approximation of p-Laplace equations, Calc. Var., 37 (2010), 485-522. 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 $\mathbbR^n$, Proc. Roy. Soc. Edinburgh Sect. A, 129 (1999), 787-809. doi: 10.1017/S0308210500013147.
[13]	B. Kawohl, Rearrangements and Convexity of Level Sets in PDE, Lect. Notes Math., 1150, Springer-Verlag, Berlin (1985).
[14]	S. Kesavan, Symmetrization and Applications, World Scientific, Hackensack, NJ, 2006.
[15]	E. Lieb and M. Loss, Analysis, Grad. Stud. Math., vol. 14, Amer. Math. Soc., Providence, RI, 2001.
[16]	J. Mawhin and M. Willen, Critical Point Theory and Hamiltonian Systems, Applied Mathematical Sciences, 74, Springer, Berlin, 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, Bull. Sci. Math., 136 (2012), 521-573. doi: 10.1016/j.bulsci.2011.12.004.
[18]	Y. Park, Fractional Polya-Zsego inequality, J. Chungcheong Math. Soc., 24 (2011), 267-271.
[19]	P. Rabinowitz, On a class of nonlinear Schrödinguer equations, ZAMP, 43 (1992), 270-291. doi: 10.1007/BF00946631.
[20]	S. Secchi, Ground state solutions for nonlinear fractional Schroinger equations in $\mathbbR^n$, J. Math. Phys., 54 (2013), 031501. 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. 187, Cambridge University Press, 2011. doi: 10.1017/CBO9780511910135.
[23]	J. Van Schaftingen, Symmetrization and minimax principle, Comm. Contemporary Math., 7 (2005), 463-481. doi: 10.1142/S0219199705001817.

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