On the logarithmic Schrodinger equation

In the framework of the nonsmooth critical point theory for lower semi-continuous functionals, we propose a direct variational approach to investigate the existence of infinitely many weak solutions for a class of semi-linear elliptic equations with logarithmic nonlinearity arising in physically relevant situations. Furthermore, we prove that there exists a unique positive solution which is radially symmetric and nondegenerate.

[1]  A. Haraux,et al.  Équations d'évolution avec non linéarité logarithmique , 1980 .

[2]  Marco Degiovanni,et al.  Euler equations involving nonlinearities without growth conditions , 1996 .

[3]  T. Cazenave.  Stable solutions of the logarithmic Schrödinger equation , 1983 .

[4]  Marion Kee,et al.  Analysis , 2004, Machine Translation.

[5]  P. Lions,et al.  Orbital stability of standing waves for some nonlinear Schrödinger equations , 1982 .

[6]  W. Rother,et al.  Nonlinear scalar field equations , 1992, Differential and Integral Equations.

[7]  J. Nieto,et al.  Global H1 solvability of the 3D logarithmic Schrödinger equation , 2010 .

[8]  Jaeyoung Byeon,et al.  Symmetry and monotonicity of least energy solutions , 2008, 0806.0299.

[9]  I. Bialynicki-Birula,et al.  Nonlinear Wave Mechanics , 1976 .

[10]  Pierre-Louis Lions,et al.  Nonlinear scalar field equations, II existence of infinitely many solutions , 1983 .

[11]  Juan Luis Vázquez,et al.  A Strong Maximum Principle for some quasilinear elliptic equations , 1984 .

[12]  Shu-Ming Chang,et al.  Spectra of Linearized Operators for NLS Solitary Waves , 2006, SIAM J. Math. Anal..

[13]  I. Bialynicki-Birula,et al.  Gaussons: Solitons of the Logarithmic Schrödinger Equation , 1979 .

[14]  Marco Degiovanni,et al.  Deformation properties for continuous functionals and critical point theory , 1993 .

[15]  L. Jeanjean,et al.  A remark on least energy solutions in RN , 2002 .

[16]  Antonio Ambrosetti,et al.  Perturbation Methods and Semilinear Elliptic Problems on R^n , 2005 .

[17]  J. Serrin,et al.  UNIQUENESS OF GROUND STATES FOR QUASILINEAR ELLIPTIC EQUATIONS IN THE EXPONENTIAL CASE , 1998 .

[18]  Marco Degiovanni,et al.  A critical point theory for nonsmooth functional , 1994 .

[19]  Marco Degiovanni,et al.  Multiple solutions of semilinear elliptic equations with one-sided growth conditions , 2000 .

[20]  Pierre-Louis Lions,et al.  Nonlinear scalar field equations, I existence of a ground state , 1983 .

[21]  Konstantin G. Zloshchastiev,et al.  Logarithmic nonlinearity in theories of quantum gravity: Origin of time and observational consequences , 2009, 0906.4282.

[22]  Manuel del Pino,et al.  The optimal Euclidean Lp-Sobolev logarithmic inequality , 2003 .

[23]  B. Gidas,et al.  Symmetry of positive solutions of nonlinear elliptic equations in R , 1981 .

[24]  Jared C. Bronski,et al.  Soliton dynamics in a potential , 2000 .

[25]  M. Squassina.  RADIAL SYMMETRY OF MINIMAX CRITICAL POINTS FOR NONSMOOTH FUNCTIONALS , 2009, 0911.1333.

[26]  E. Lieb,et al.  Analysis, Second edition , 2001 .

[27]  M. Shubin,et al.  The Schrödinger Equation , 1991 .

[28]  Marco Degiovanni,et al.  Subdifferential Calculus and Nonsmooth Critical Point Theory , 2000, SIAM J. Optim..

[29]  Michael Struwe,et al.  Variational methods: Applications to nonlinear partial differential equations and Hamiltonian systems , 1990 .