Existence of bound and ground states for an elliptic system with double criticality

We study the existence of bound and ground states for a class of nonlinear elliptic systems in R . These equations involve critical power nonlinearities and Hardy-type singular potentials, coupled by a term containing up to critical powers. More precisely, we find ground states either the positive coupling parameter ν is large or ν is small under suitable assumptions on the other parameters of the problem. Furthermore, bound states are found as Mountain–Pass-type critical points of the underlying functional constrained on the Nehari manifold. Our variational approach improves some known results and allows us to cover range of parameters which have not been considered previously.

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

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

[3]  J. Rossi,et al.  The critical hyperbola for a Hamiltonian elliptic system with weights , 2008 .

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

[5]  E. Colorado.  On the existence of bound and ground states for some coupled nonlinear Schrödinger–Korteweg–de Vries equations , 2014, 1411.7283.

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

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

[8]  David Ruiz,et al.  Multi-bump solitons to linearly coupled systems of nonlinear Schrödinger equations , 2007 .

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

[10]  J. Pava,et al.  Existence and Stability of Ground-state Solutions of a Schrödinger-kdv System , 2022 .

[11]  Eduardo Colorado,et al.  Bound and ground states of coupled “NLS-KdV” equations with Hardy potential and critical power , 2021, Journal of Differential Equations.

[12]  J. Dias,et al.  Well-posedness and existence of bound states for a coupled Schrödinger-gKdV system , 2010 .

[13]  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.

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

[15]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[16]  Zhaoli Liu,et al.  Ground States and Bound States of a Nonlinear Schrödinger System , 2010 .

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

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

[19]  Thomas,et al.  NOTE ON GROUND STATES OF NONLINEAR SCHRODINGER SYSTEMS , 2006 .

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

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

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

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

[24]  Wenming Zou,et al.  Existence and symmetry of positive ground states for a doubly critical Schrodinger system , 2012, 1211.1514.

[25]  Funakoshi Mitsuaki,et al.  The Resonant Interaction between a Long Internal Gravity Wave and a Surface Gravity Wave Packet , 1983 .

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