An existence theory for nonlinear equations on metric graphs via energy methods

2019/09/17 by Matthias Hofmann, Hofmann, Matthias · 2 citations
Computer Science · Mathematics · #35J20 #35J35 #35J60 #35Q55 #35R02 (35J10 #49J40 #81Q35) #Advanced Mathematical Modeling in Engineering #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.1909.07856

openalex publication_date 2019/09/17 · openalex created_date 2019/09/26 · openalex updated_date 2026/07/28

Abstract

The purpose of this paper is to develop a general existence theory for constrained minimization problems for functionals defined on function spaces on metric measure spaces (\mathcal M, d, μ). We apply this theory to functionals defined on metric graphs \mathcal G, in particular L²-constrained minimization problems for functionals of the form E(u) = (1)/(2) a(u,u) - (1)/(q)∫_{\mathcal K} |u|^q \mathrm dx, where q>2, a(⋅, ⋅) is a suitable symmetric sesquilinear form on some function space on \mathcal G and \mathcal K ⊆ \mathcal G is given. We show how the existence of solutions can be obtained via decomposition methods using spectral properties of the operator A associated with the form a(⋅, ⋅) and discuss the spectral quantities involved. An example that we consider is the higher-order variant of the stationary NLS (nonlinear Schrödinger) energy functional with potential V∈ L²+ L^∞(\mathcal G) E^(k)(u)= (1)/(2) ∫_{\mathcal G} |u^(k)|²+ V(x) |u|² \mathrm dx - (1)/(p) ∫_{\mathcal K} |u|^q \mathrm dx defined on a class of higher-order Sobolev spaces H^k(\mathcal G) that we introduce. When \mathcal K is a bounded subgraph, one has localized nonlinearities, which we treat as a special case. When k=1 we also consider metric graphs with infinite edge set as well as magnetic potentials. Then the operator A associated to the linear form is a Schrödinger operator, and in the L²-subcritical case 2

Citations

• A relation between pointwise convergence of functions and convergence of functionals

Cited by

• Normalized solutions of nonlinear magnetic Schrödinger equations on metric graphs
• Spectral minimal partitions of unbounded domains

Related

• Existence of infinite stationary solutions of the L²-subcritical and critical NLSE on compact metric graphs
• Multiple positive bound states for the subcritical NLS equation on metric graphs
• Local minimizers in absence of ground states for the critical NLS energy\n on metric graphs
• On the notion of ground state for nonlinear Schrödinger equations on metric graphs
• Variational and stability properties of constant solutions to the NLS\n equation on compact metric graphs