NODAL BOUND STATES WITH CLUSTERED SPIKES FOR NONLINEAR SCHRÖDINGER EQUATIONS

Abstract We consider the following nonlinear Schrodinger equations - ɛ 2 Δ u + u = Q ( x ) | u | p - 2 u i n ℝ N , u ∈ H 1 ( ℝ N ) , where ɛ is a small positive parameter, N ≥ 2 , 2 p ∞ for N = 2 and 2 p 2 N N - 2 for N ≥ 3. We prove that this problem has sign-changing (nodal) semi-classical bound states with clustered spikes for sufficiently small ɛ under some additional conditions on Q ( x ). Moreover, the number of this type of solutions will go to infinity as ɛ → 0 + .