Nonlinear Schrodinger equations with steep potential well

We investigate nonlinear Schrodinger equations like the model equation -Deltau + V-lambda(x)u = \u \ (p-2)u, x is an element of R-N, 2 < p < 2*, where the potential V-lambda has a potential well with bottom independent of the parameter lambda > 0. If lambda --> infinity the infimum of the essential spectrum of -Delta + V-lambda in L-2(R-N) converges towards oo and more and more eigenvalues appear below the essential spectrum. We show that as lambda --> infinity more and more solutions of the nonlinear Schrodinger equation exist. The solutions lie in H-1(R-N) and are localized near the bottom of the potential well, but not near local minima of the potential. We also investigate the decay rate of the solutions as \x \ --> infinity as well as their behaviour as lambda --> infinity.