Groundstates for the nonlinear Schrodinger equation with potential vanishing at infinity

Bonheure, Denis;Van Schaftingen, Jean
(2010) Annali di Matematica Pura ed Applicata — Vol. 189, n° 2, p. 273-301 (2010)

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Abstract
Groundstates of the stationary nonlinear Schrodinger equation -Delta u + Vu = Kup(-1) are studied when the nonnegative function V and K are neither bounded away from zero, nor bounded from above. A special attention is paid in the case of a potential V that goes to 0 at infinity. Conditions on compact embeddings that allow to prove in particular the existence of groundstates are established. The fact that the solution is in L-2(R-N) is studied and decay estimates are derived using Moser iteration scheme. The results depend on whether V decays slower than vertical bar x vertical bar(-2) at infinity.
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Bonheure, D., & Van Schaftingen, J. (2010). Groundstates for the nonlinear Schrodinger equation with potential vanishing at infinity. Annali di Matematica Pura ed Applicata, 189(2), 273-301. https://doi.org/10.1007/s10231-009-0109-6 (Original work published 2010)