Geodesic
Ground states of nonlinear Schrödinger equations with potentials vanishing at infinity
Journal of the European Mathematical Society, Tome 7 (2005) no. 1, pp. 117-144.

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We deal with a class on nonlinear Schrödinger equations (NLS) with potentials V(x)∼∣x∣−α, 02, and K(x)∼∣x∣−β, β>0. Working in weighted Sobolev spaces, the existence of ground states vε​ belonging to W1,2(Rn) is proved under the assumption that σ(N+2)/(N−2) for some σ=σN,α,β​. Furthermore, it is shown that vε​ are spikes concentrating at a minimum of A=VθK−2/(p−1), where θ=(p+1)/(p−1)−1/2.
DOI : 10.4171/jems/24
Classification : 35-XX, 00-XX
Keywords: Nonlinear Schrödinger equations, weighted Sobolev spaces 
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     title = {Ground states of nonlinear {Schr\"odinger} equations with potentials vanishing at infinity},
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Antonio Ambrosetti; Veronica Felli; Andrea Malchiodi. Ground states of nonlinear Schrödinger equations with potentials vanishing at infinity. Journal of the European Mathematical Society, Tome 7 (2005) no. 1, pp. 117-144. doi : 10.4171/jems/24. http://geodesic.mathdoc.fr/articles/10.4171/jems/24/

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