Groundstates for the nonlinear Schrödinger equation with potential vanishing at infinity

AbstractGroundstates of the stationary nonlinear Schrödinger equation $$-\Delta u +V u =K u^{p-1}$$, 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 $${L^2(\mathbb R^N)}$$ 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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