Abstract
Groundstates of the stationary nonlinear Schrödinger equation
, 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 \({L^2(\mathbb R^N)}\) is studied and decay estimates are derived using Moser iteration scheme. The results depend on whether V decays slower than |x|−2 at infinity.
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Both authors were supported by the Fonds de la Recherche Scientifique, FNRS (Communauté française de Belgique). JVS was supported by the Fonds spéciaux de recherche (Université catholique de Louvain).
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Bonheure, D., Van Schaftingen, J. Groundstates for the nonlinear Schrödinger equation with potential vanishing at infinity. Annali di Matematica 189, 273–301 (2010). https://doi.org/10.1007/s10231-009-0109-6
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DOI: https://doi.org/10.1007/s10231-009-0109-6
Keywords
- Stationary nonlinear Schrödinger equation
- Decay of solutions
- Weighted Sobolev spaces
- Regularity theory
- Moser iteration scheme
- Trace inequalities
- Degenerate potentials