On a class of nonlinear inhomogeneous Schrödinger equation

Abstract

In this paper, we study the inhomogeneous Schrödinger equationBy using variational methods and a refined interpolation inequality, we establish some simple but sharp conditions on the solutions which exist globally or blow up in a finite time. An interesting result is that we obtain the existence of global solution for arbitrarily large data.

Global well-posedness and blow-up on the energy space for the inhomogeneous nonlinear Schrödinger equation

“…However, in both papers, the authors assume that k(x) is bounded which is not verified in our case. Another type of the inhomogeneous nonlinear Schrödinger equation (INLS), but with nonlinearity of the form |x| b |u| 2σ u with b > 0, was studied by Chen and Guo [4] and Chen [3]. In these papers the authors obtain certain conditions for global existence and blow-up in the set of radial symmetric functions in H 1 (R N ).…”

Blow-up rate, mass concentration and asymptotic profile of blow-up solutions for the nonlinear inhomogeneous Schrödinger equation

“…For the nonlinearity with unbounded potential jxj b , Chen and Guo [3] established the local well-posedness of the Cauchy problem (1.1), (1.2) in H 1 r ¼ H 1 r ðR N Þ, where H 1 r ðR N Þ is the set of radial symmetric functions in H 1 ðR N Þ. Chen and Guo [3], Chen [4] studied the existence of blow-up solutions. Due to the unbounded potential jxj b , to our knowledge, there are few results about blow-up solutions for Cauchy problem (1.1), (1.2), which motivates us to do further research on the dynamics of blowup solutions for the Cauchy problem (1.1), (1.2 …”

“…The minimal energy solution RðxÞ of (1.6) is called the ground state solution (see [2,4]), and the general solution of (1.6) is called the bound state solution. Sintzoff and Willem [15] proved the existence of bound state solutions of (1.6).…”

“…Sintzoff and Willem [15] proved the existence of bound state solutions of (1.6). Chen [4] studied the existence of the ground state solution of (1.6), and gave a sharp generalized Gagliardo-Nirenberg inequality. Then, Chen [4] obtained the sharp threshold of blow-up and global existence for Cauchy problem (1.1), (1.2).…”

“…Chen [4] studied the existence of the ground state solution of (1.6), and gave a sharp generalized Gagliardo-Nirenberg inequality. Then, Chen [4] obtained the sharp threshold of blow-up and global existence for Cauchy problem (1.1), (1.2). It reads that if the initial data ku 0 k L 2 < kRk L 2 , then the solution uðt; xÞ exists globally; if the initial data ku 0 k L 2 P kRk L 2 , then the solution uðt; xÞ may blow up, which implies that kRk L 2 is the minimal mass of the existence of blow-up solutions.…”