A type of quasilinear Schrödinger equations in two space dimensions which describe attractive Bose-Einstein condensates in physics is discussed. By establishing the property of the equation and applying the energy method, the blowup of solutions to the equation are proved under certain conditions. At the same time, by the variational method, a sufficient condition of global existence which is related to the ground state of a classical elliptic equation is obtained.
quasilinear Schrödinger equations blowup global existence ground state Bose-Einstein condensates
Chinese Library Classification
2000 Mathematics Subject Classification
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Poppenberg M. An inverse function theorem in Sobolev spaces and applications to quasi-linear Schrödinger equations[J]. J Math Anal Appl, 2001, 258(1):146–170.zbMATHCrossRefMathSciNetGoogle Scholar
Poppenberg M, Schmitt K, Wang Z Q. On the existence of soliton solutions to quasi-linear Schrödinger equations[J]. Calc Var Partial Differential Equations, 2002, 14(3):329–344.zbMATHCrossRefMathSciNetGoogle Scholar
Liu J Q, Wang Y Q, Wang Z Q. Soliton solutions for quasi-linear Schrödinger equations II[J]. J Diff Eq, 2003, 187(3):473–493.zbMATHCrossRefGoogle Scholar
García-Ripoll Juan J, Konotop Vladimit V, Malomed Boris, Perez-Garca V M. A quasi-local Gross-Pitaevskii equation for attractive Bose-Einstein condensates[J]. Mathematics and Computers in Simulation, 2003, 62(1/2):21–30.zbMATHCrossRefMathSciNetGoogle Scholar
Zhang Jian. The blowup properties of the initial-boundary problem for second order derivative nonlinear Schrödinger equations[J]. Acta Mathematica Scientia, 1994, 14(suppl):89–94 (in Chinese).Google Scholar