Article Contents
Article Contents

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

• 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.
Mathematics Subject Classification: Primary: 35J60, 35R11; Secondary: 35J20, 35B07.

 Citation:

•  [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 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.