Semiclassical stationary states for nonlinear Schrodinger equations with fast decaying potentials

Moroz, Vitaly;Van Schaftingen, Jean
(2010) Calculus of Variations and Partial Differential Equations — Vol. 37, n° 1-2, p. 1-27 (2010)

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Abstract
We study the existence of positive solutions for a class of nonlinear Schrodinger equations of the type -epsilon(2)Delta u + Vu = u(p) in R-N, where N >= 3, p > 1 is subcritical and V is a nonnegative continuous potential. Amongst other results, we prove that if V has a positive local minimum, and N/N-2 < p < N+2/N-2 , then for small epsilon the problem admits positive solutions which concentrate as epsilon -> 0 around the local minimum point of V. The novelty is that no restriction is imposed on the rate of decay of V. In particular, we cover the case where V is compactly supported.
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Moroz, V., & Van Schaftingen, J. (2010). Semiclassical stationary states for nonlinear Schrodinger equations with fast decaying potentials. Calculus of Variations and Partial Differential Equations, 37(1-2), 1-27. https://doi.org/10.1007/s00526-009-0249-y (Original work published 2010)