Blow-up phenomena and asymptotic profiles passing from H¹-critical to super-critical quasilinear Schrödinger equations

2021/04/14 by Daniele Cassani, Cassani, Daniele, Youjun Wang +1
Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2104.06816

openalex publication_date 2021/04/14 · openalex created_date 2021/04/26 · openalex updated_date 2026/07/28

Abstract

We study the asymptotic profile, as ℏ→ 0, of positive solutions to -ℏ²Δu+V(x)u-ℏ^2+γuΔu²=K(x)|u|^p-2u, x∈ ℝ^N where γ≥ 0 is a parameter with relevant physical interpretations, V and K are given potentials and N≥ 5. We investigate the concentrating behavior of solutions when γ>0 and, differently form the case γ=0 where the leading potential is V, the concentration is here localized by the source potential K. Moreover, surprisingly for γ>0 we find a different concentration behavior of solutions in the case p=(2N)/(N-2) and when (2N)/(N-2)

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