Resumen de Positive solutions for nonlinear Schrödinger equations with deepening potential well

Zhengping Wang, Huan-Song Zhou

• Consider the following nonlinear Schrödinger equation:

  (*) -?u + (1 + ?g(x))u = f(u) and u> 0 in RN, u ? H1.(RN), N = 3, where ? = 0 is a parameter, g ? L8(RN) vanishes on a bounded domain in RN, and the function f is such that lim(s?0) f(s)/s = 0 and 1 = a + 1 = lim(s?8) f(s)/s < 8.

  We are interested in whether problem (*) has a solution for any given a, ? = 0. It is shown in [14] and [31] that problem (*) has solutions for some a and ?. In this paper, we establish the existence of solution of (*) for all a and ? by using a variant of the Mountain Pass Theorem. Based on these results, we give a diagram in the ("l",a)-plane showing how the solvability of problem (*) depends on the parameters a and "l".