Bifurcation into spectral gaps for strongly indefinite Choquard equations

2022/05/05 by Huxiao Luo, Luo, Huxiao, Bernhard Ruf +3
Computer Science · Engineering · Mathematics · #Advanced Mathematical Modeling in Engineering #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.2205.02542

openalex publication_date 2022/05/05 · openalex created_date 2023/02/15 · openalex updated_date 2026/07/28

Abstract

We consider the semilinear elliptic equations \ · amp;-Δu+V(x)u=(I_α∗ |u|^p)|u|^p-2u+λu \hboxfor x∈\mathbb R^N,
· amp;u(x) → 0 \hbox as |x| →∞, . where I_α is a Riesz potential, p∈(\fracN+αN,(N+α)/(N-2)), N≥3, and V is continuous periodic. We assume that 0 lies in the spectral gap (a,b) of -Δ+ V. We prove the existence of infinitely many geometrically distinct solutions in H¹(\mathbb R^N) for each λ∈(a, b), which bifurcate from b if \fracN+αN< p < 1 +(2+α)/(N). Moreover, b is the unique gap-bifurcation point (from zero) in [a,b]. When λ=a, we find infinitely many geometrically distinct solutions in H²_loc(\mathbb R^N). Final remarks are given about the eventual occurrence of a bifurcation from infinity in λ=a.

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