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The purpose of this paper is to develop a general existence theory forconstrained minimization problems for functionals defined on function spaces onmetric measure spaces $(\mathcal M, d, \mu)$. We apply this theory tofunctionals defined on metric graphs $\mathcal G$, in particular$L^2$-constrained minimization problems for functionals of the form $$E(u) =\frac{1}{2} a(u,u) - \frac{1}{q}\int_{\mathcal K} |u|^q \, \mathrm dx,$$ where$q>2$, $a(\cdot, \cdot)$ is a suitable symmetric sesquilinear form on somefunction space on $\mathcal G$ and $\mathcal K \subseteq \mathcal G$ is given.  We show how the existence of solutions can be obtained via decompositionmethods using spectral properties of the operator $A$ associated with the form$a(\cdot, \cdot)$ and discuss the spectral quantities involved. An example thatwe consider is the higher-order variant of the stationary NLS (nonlinearSchr\"odinger) energy functional with potential $V\in L^2+ L^\infty(\mathcalG)$ $$ E^{(k)}(u)= \frac{1}{2} \int_{\mathcal G} |u^{(k)}|^2+ V(x) |u|^2 \,\mathrm dx - \frac{1}{p} \int_{\mathcal K} |u|^q \, \mathrm dx$$ defined on aclass of higher-order Sobolev spaces $H^k(\mathcal G)$ that we introduce. When$\mathcal K$ is a bounded subgraph, one has localized nonlinearities, which wetreat as a special case. When $k=1$ we also consider metric graphs withinfinite edge set as well as magnetic potentials. Then the operator $A$associated to the linear form is a Schr\"odinger operator, and in the$L^2$-subcritical case $2<q<6$, we obtain generalizations of existence resultsfor the NLS functional as for instance obtained by Adami, Serra and Tilli [JFA271 (2016), 201-223], and Cacciapuoti, Finco and Noja [Nonlinearity 30 (2017),3271-3303], among others.

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