Global well-posedness, scattering and blow-up for the energy-critical, Schrödinger equation with indefinite potential in the radial case

2024/06/22 by Jun Wang, Zhaoyang Yin, Wang, Jun +1
Mathematics · #35Q55 #35R11 #37K05 #37L50 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.2407.01588

openalex publication_date 2024/06/22 · openalex created_date 2024/07/06 · openalex updated_date 2026/07/28

Abstract

In this paper, we study the well-posedness theory and the scattering asymptotics for the energy-critical, Schrödinger equation with indefinite potential \i ∂_t u+Δu-V(x)u +|u|^(4)/(N-2)u=0, (x, t) ∈ ℝ^N × ℝ,
.u|_t=0=u₀ ∈ H ¹(ℝ^N),. where V(x):ℝ^N→ ℝ is indefinite and satisfies appropriate conditions. Using contraction mapping method and concentration compactness argument, we obtain the well-posedness theory in proper function spaces and scattering asymptotics. Moreover, we get a positive ground state solution which is radially symmetric by using variational methods. This paper extends the results of \citeKCEMF2006(Invent. Math) to the potential equation and develops the recent conclusions.

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