Existence and symmetry of positive ground states for a doubly critical Schrodinger system

N N−2 and α > 1,β > 1 satisfying α+β = 2 ∗ . This problem is related to coupled nonlinear Schrodinger equations with critical exponent for Bose-Einstein condensate. For different ranges of N, α, β and ν > 0, we obtain positive ground state solutions via some quite different methods, which are all radially symmetric. It turns out that the least energy level depends heavily on the relations among α, β and 2. Besides, for sufficiently smallν > 0, positive solutions are also obtained via a variational perturbation approach. Note that the Palais-Smale condition can not hold for any positive energy level, which makes the study via variational methods rather complicated.

[1]  Adrian Ankiewicz,et al.  PARTIALLY COHERENT SOLITONS ON A FINITE BACKGROUND , 1999 .

[2]  Susanna Terracini,et al.  On positive entire solutions to a class of equations with a singular coefficient and critical exponent , 1996, Advances in Differential Equations.

[3]  Zhaoli Liu,et al.  Multiple Bound States of Nonlinear Schrödinger Systems , 2008 .

[4]  D. J. Frantzeskakis,et al.  Dark solitons in atomic Bose–Einstein condensates: from theory to experiments , 2010, 1004.4071.

[5]  Pierre-Louis Lions,et al.  The concentration-compactness principle in the Calculus of Variations , 1985 .

[6]  V. Felli,et al.  Existence and multiplicity for perturbations of an equation involving a Hardy inequality and the critical Sobolev exponent in the whole of $\Bbb R^N$ , 2004, Advances in Differential Equations.

[7]  Chris H. Greene,et al.  Hartree-Fock Theory for Double Condensates , 1997 .

[8]  B. Gidas,et al.  Symmetry and related properties via the maximum principle , 1979 .

[9]  Antonio Ambrosetti,et al.  Standing waves of some coupled nonlinear Schrödinger equations , 2007 .

[10]  V. Felli,et al.  Existence of Blowing-up Solutions for a Nonlinear Elliptic Equation with Hardy Potential and Critical Growth , 2006 .

[11]  Yuri S. Kivshar,et al.  Dark optical solitons: physics and applications , 1998 .

[12]  Michael Struwe,et al.  A global compactness result for elliptic boundary value problems involving limiting nonlinearities , 1984 .

[13]  Haim Brezis,et al.  Positive Solutions of Nonlinear Elliptic Equations Involving Critical Sobolev Exponents Haim Brezis , 2022 .

[14]  B. Sirakov.  Least Energy Solitary Waves for a System of Nonlinear Schrödinger Equations in $${\mathbb{R}^n}$$ , 2007 .

[15]  Wenxiong Chen,et al.  Classification of solutions for an integral equation , 2006 .

[16]  Tai-Chia Lin,et al.  Ground State of N Coupled Nonlinear Schrödinger Equations in Rn,n≤3 , 2005 .

[17]  Elliott H. Lieb,et al.  A Relation Between Pointwise Convergence of Functions and Convergence of Functionals , 1983 .

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

[19]  G. Talenti,et al.  Best constant in Sobolev inequality , 1976 .

[20]  J. R. Ensher,et al.  Dynamics of component separation in a binary mixture of Bose-Einstein condensates , 1998 .

[21]  D. Smets.  Nonlinear Schrödinger equations with Hardy potential and critical nonlinearities , 2005 .

[22]  Jaeyoung Byeon,et al.  Standing Waves for Nonlinear Schrödinger Equations with a General Nonlinearity , 2007 .

[23]  Coupled nonlinear Schrödinger systems with potentials , 2005, math/0506010.

[24]  M. Willem.  Minimax Theorems , 1997 .

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

[26]  P. Lions.  The concentration-compactness principle in the Calculus of Variations , 1984 .

[27]  P. Rabinowitz,et al.  Dual variational methods in critical point theory and applications , 1973 .

[28]  T. Aubin,et al.  Problèmes isopérimétriques et espaces de Sobolev , 1976 .

[29]  Veronica Felli,et al.  Some remarks on systems of elliptic equations doubly critical in the whole $${\mathbb{R}^N}$$ , 2009 .

[30]  W. Zou,et al.  Positive Least Energy Solutions and Phase Separation for Coupled Schrödinger Equations with Critical Exponent , 2012, 1209.2522.

[31]  Zoubin Ghahramani,et al.  Variational Methods , 2014, Computer Vision, A Reference Guide.

[32]  B. Pellacci,et al.  Positive solutions for a weakly coupled nonlinear Schrödinger system , 2006 .

[33]  S. Terracini,et al.  Multipulse Phases in k-Mixtures of Bose–Einstein Condensates , 2008, 0807.1979.