Abstract
We study the existence of positive solutions for a class of nonlinear Schrödinger equations of the type
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 \({\frac{N}{N-2} < p < \frac{N+2}{N-2}}\) , then for small ε the problem admits positive solutions which concentrate as ε → 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. Semiclassical stationary states for nonlinear Schrödinger equations with fast decaying potentials. Calc. Var. 37, 1–27 (2010). https://doi.org/10.1007/s00526-009-0249-y
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DOI: https://doi.org/10.1007/s00526-009-0249-y