American Institute of Mathematical Sciences

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

 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.
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 $\mathbb{R}^{N}$, 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 $\mathbb{R}^{N}$, J. Math. Phys., 54 (2013), 031501. doi: 10.1063/1.4793990. [21] S. Secchi, On fractional Schrödinger equation in $\mathbb{R}^{N}$ without the Ambrosetti-Rabinowitz condition, to appear in 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 $\mathbb{R}^{N}$, 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 $\mathbb{R}^{N}$, J. Math. Phys., 54 (2013), 031501. doi: 10.1063/1.4793990. [21] S. Secchi, On fractional Schrödinger equation in $\mathbb{R}^{N}$ without the Ambrosetti-Rabinowitz condition, to appear in 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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