2014/04/30 by Przemysław Zieliński, Zieliński, Przemysław
Engineering · Mathematics · #35J10 (Secondary) #46N20 #47H05 (Primary) 47J05 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Functional Analysis (math.FA) #Spectral Theory in Mathematical Physics #Stability and Controllability of Differential Equations #math.AP #math.FA #msc:35J10 #msc:46N20 #msc:47H05 #msc:47J05
paper · pdf · doi:10.48550/arxiv.1404.7624
18 pages
arxiv created 2014/04/30 · openalex publication_date 2014/04/30 · arxiv updated 2014/05/01 · openalex created_date 2022/08/28 · openalex updated_date 2026/07/28
We study the existence of solutions in Hilbert space H of the semilinear equation L u+N(u)=h, where L is linear self-adjoint, N is a nonlinear operator and h∈ H. We concentrate on the case when 0 is a right boundary point of a gap in the spectrum of L and an element of essential spectrum. The sufficient conditions for solvability are based on monotonicity and sign assumptions on operator N, and its behaviour on ker L. We illustrate the main theorem by an application to the study of nonlinear stationary Schrödinger equation on ℝn.