NONLINEAR SCHRÖDINGER EQUATIONS WITH STEEP POTENTIAL WELL

We investigate nonlinear Schrodinger equations like the model equation \[ -\Delta u +V_\lambda (x) u =| u |^{p -2} u\,, \quad x\in {\mathbb R}^N\,, \ 2 0. If λ → ∞ the infimum of the essential spectrum of -Δ + Vλ in L2(ℝN) converges towards ∞ and more and more eigenvalues appear below the essential spectrum. We show that as λ→∞ more and more solutions of the nonlinear Schrodinger equation exist. The solutions lie in H1(ℝN) 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→∞.