Universal critical power for nonlinear Schrodinger equations with symmetric double well potential

Here we consider stationary states for nonlinear Schrodinger equations with symmetric double well potentials. These stationary states may bifurcate as the strength of the nonlinear term increases and we observe two different pictures depending on the value of the nonlinearity power: a simple pitch-fork bifurcation, and a couple of saddle points which unstable branches collapse in an inverse pitch-fork bifurcation. In this paper we show that in the semiclassical limit, or when the barrier between the two wells is large enough, the first kind of bifurcation always occurs when the nonlinearity power is less than a critical value (2); in contrast, when the nonlinearity power is larger than such a critical value then we always observe the second scenario. The remarkable fact is that such a critical value is an universal constant in the sense that it does not depends on the shape of the double well potential.