CONCENTRATION BEHAVIOR OF GROUND STATES FOR L2 -CRITICAL SCHRODINGER EQUATION WITH A SPATIALLY DECAYING NONLINEARITY

作者：Luo, Yong; Zhang, Shu^*

来源：Communications on Pure and Applied Analysis, 2022, 21(4): 1481-1504.

DOI：10.3934/cpaa.2022026

• ground states
• concentration behavior
• local uniqueness
• constrained variational method
• L-2-critical

We consider ground states of the following time-independent nonlinear L-2-critical Schriidinger equation @@@ -Delta u(x) + V(x)u(x) - a vertical bar x vertical bar(-b)vertical bar u vertical bar(4-2b/N) u(x) = mu u(x) in R-N, @@@ where mu is an element of R, a > 0, N >= 1, 0 < b < min{2, N}, and V(x) >= 0 is an external potential. We get ground states of the above equation by solving the associated constrained minimization problem. In this paper, we prove that there is a threshold a* >0 such that minimizer exists for 0 < a < a*, and minimizer does not exist for any a> a*. However if a = a* , it is showed that whether minimizer exists depends sensitively on the value of V(0). Moreover if V(0) =0, we prove that minimizers must concentrate at the origin as a NE arrow a* and give a detailed concentration behavior of minimizers as a NE arrow a*, based on which we finally prove that there is a unique minimizer when a is close enough to a*.