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Ground states of nonlinear Schrödinger equations with potentials vanishing at infinity

Antonio AmbrosettiVeronica FelliAndrea Malchiodi — 2005

Journal of the European Mathematical Society

We deal with a class on nonlinear Schrödinger equations (NLS) with potentials V ( x ) | x | α , 0 < α < 2 , and K ( x ) | x | β , β > 0 . Working in weighted Sobolev spaces, the existence of ground states v ε belonging to W 1 , 2 ( N ) is proved under the assumption that σ < p < ( N + 2 ) / ( N 2 ) for some σ = σ N , α , β . Furthermore, it is shown that v ε are spikes concentrating at a minimum point of 𝒜 = V θ K 2 / ( p 1 ) , where θ = ( p + 1 ) / ( p 1 ) 1 / 2 .

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