The existence of solutions for a Schrodinger equation with jumping nonlinearities crossing the essential spectrum

2024/07/04 by Chong Li, Xinyu Li, Li, Chong +1
Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2407.03717

openalex publication_date 2024/07/04 · openalex created_date 2024/07/09 · openalex updated_date 2026/07/28

Abstract

In this paper, we establish the existence of one solution for a Schrödinger equation with jumping nonlinearities: -Δu+V(x)u=f(x,u), x∈ \mathbb R^N, and u(x)→ 0, |x|→ +∞, where V is a potential function on which we make hypotheses, and in particular allow V which is unbounded below, and f(x,u)=au^-+bu⁺+g(x,u). No restriction on b is required, which implies that f(x,s)s^-1 may interfere with the essential spectrum of -Δ+V for s→ +∞. Using the truncation method and the Morse theory, we can compute critical groups of the corresponding functional at zero and infinity, then prove the existence of one negative solution.

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