Nonlinear Schrödinger Equations With Steep Potential Well

We investigate nonlinear Schrödinger equations like the model equation [Formula: see text] where the potential V λ has a potential well with bottom independent of the parameter λ > 0. If λ → ∞ the infimum of the essential spectrum of -Δ + V λ in L 2 (ℝ N ) converges towards ∞ and more and more eigenvalues appear below the essential spectrum. We show that as λ→∞ more and more solutions of the nonlinear Schrödinger 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|→∞ as well as their behaviour as λ→∞.