On Ambrosetti-Malchiodi-Ni Conjecture for General Hypersurfaces

We consider the nonlinear problem where p > 1, ε is a small parameter and V is a uniformly positive, smooth potential. Assume that R ⊂ R ⁿis a smooth closed, stationary and non-degenerate hypersurface relative to the functional ∫ _RV ^Σ withΣ=P+1/P-1-1/2. We prove the existence of solutions,ũ _ε at least for some sequence {ε _l} _l which concentrate along smooth surfaces T _ε close to R This result confirms the validity of the conjecture of Ambrosetti et al. in [2] for concentration of Schrödinger equation on general hypersurfaces.