Multiplicity and concentration results for local and fractional NLS equations with critical growth

2021/01/02 by Gallo, Marco
#35A15 #35B25 #35B33 #35Q55 #35R11 #47J30 #58E05 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2101.00448

Abstract

Goal of this paper is to study positive semiclassical solutions of the nonlinear Schrödinger equation ε^2s(- Δ)^s u+ V(x) u= f(u), x ∈ ℝ^N, where s ∈ (0,1), N ≥ 2, V ∈ C(ℝ^N,ℝ) is a positive potential and f is assumed critical and satisfying general Berestycki-Lions type conditions. We obtain existence and multiplicity for ε>0 small, where the number of solutions is related to the cup-length of a set of local minima of V. Furthermore, these solutions are proved to concentrate in the potential well, exhibiting a polynomial decay. We highlight that these results are new also in the limiting local setting s=1 and N≥ 3, with an exponential decay of the solutions.

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