Local uniqueness and the number of concentrated solutions for nonlinear Schrodinger equations with non-admissible potential

作者：Luo, Peng^*; Tian, Shuying; Zhou, Xiaodong

来源：Nonlinearity, 2021, 34(2): 705-724.

DOI：10.1088/1361-6544/abd1cb

• nonlinear Schrodinger equation
• non-isolated critical points
• uniqueness

We revisit the following nonlinear Schrodinger equation -epsilon(2)Delta u+V(x)u = u(p1), u > 0, u is an element of H-1(R-N), where epsilon > 0 is a small parameter, N >= 2, 2 < p < 2*. Here we are mainly concerned with local uniqueness and the number of concentrated solutions of the above nonlinear Schrodinger equation for a kind of non-admissible potential V(x) which possesses non-isolated critical points. First, we establish a more accurate location for the concentrated points, which will need an observation on the structure of the potential. Next, we prove local uniqueness for positive single-peak solutions. Then some results concerning on the number and symmetry of single-peak solutions are also given. To our knowledge, this seems to be the first result on local uniqueness and the number of concentrated solutions with non-admissible potential, which generalizes Grossi's results (2002 Inst. H. Poincare Anal. Non Lineaire 19 261-80)

• 单位
  武汉理工大学