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

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 $(\mathcal{M},d,\mu )$. We apply this theory to functionals defined on metric graphs $\mathcal{G}$, in particular $L^2$-constrained minimization problems for functionals of the form $$E(u)=\frac{1}{2}a(u,u)-\frac{1}{q}{\int }_{\mathcal{K}}|u{|}^q\mathrm{d}x,$$ where $q>2$, $a(\cdot ,\cdot )$ is a suitable symmetric sesquilinear form on some function space on $\mathcal{G}$ and $\mathcal{K}\subseteq \mathcal{G}$ 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 $a(\cdot ,\cdot )$ 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 $V\in L^2+L^{\infty }(\mathcal{G})$ $$E^{(k)}(u)=\frac{1}{2}{\int }_{\mathcal{G}}|u^{(k)}{|}^2+V(x)|u{|}^2\mathrm{d}x-\frac{1}{p}{\int }_{\mathcal{K}}|u{|}^q\mathrm{d}x$$ 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 $L^2$-subcritical case $2<q<6$, we obtain generalizations of existence results for the NLS functional as for instance obtained by Adami, Serra and Tilli [JFA 271 (2016), 201-223], and Cacciapuoti, Finco and Noja [Nonlinearity 30 (2017), 3271-3303], among others.

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Cited by in corpus (1)

• Symmetry breaking in two-dimensional square grids: persistence and failure of the dimensional crossover