On the symmetry of the ground states of nonlinear schrödinger equation with potential

We investigate the minimizers of the energy functional ε(u) = 1/2 /ℝN|δu|²dx+ 1/2/ℝN V|u|²dx-1/P+1/ℝN b|u|p+1dx under the constraint of the L²-norm. We show that for the case L²-norm is small, the minimizer is unique and for the case L²-norm is large, the minimizer concentrate at the maximum point of 6 and decays exponentially. By this result, we can show that if V and 6 are radially symmetric but 6 does not attain its maximum at the origin, then the symmetry breaking occurs as the L²-norm increases. Further, we show that for the case 6 has several maximum points, the minimizer concentrates at a point which minimizes a function which is defined by b, V and the unique positive radial solution of -δPdbl; + Pdbl; -ℙ^p =0. For the case when V and b are radially symmetric, we show that if the minimizer concentrates at the origin, then the minimizer is radially symmetric. Further, we construct an energy functional such that the minimizer breaks its symmetry once but after that it recovers to be symmetric as the L²-norm increases.