Normalized solutions of \(L^2\)-supercritical NLS equations on compact metric graphs (Q6569711)

Let \({\mathcal{G}}=({\mathcal{E}},{\mathcal{V}})\) be a compact metric graph. The authors show that if \(p>6\), then there exists \(\mu_1\in]0,+\infty[\) such that for any \(\mu\in]0,\mu_1[\), the energy functional\N\[\NE(u,{\mathcal{G}})=\frac{1}{2}\int_{ {\mathcal{G}} }|u'|^2\,dx-\frac{1}{p}\int_{ {\mathcal{G}} }|u|^p\,dx\N\]\Nunder the mass constraint \(\int_{ {\mathcal{G}} }|u|^2\,dx=\mu\) has a positive nonconstant critical point \(u\in H^1({\mathcal{G}} )\) at a strictly larger energy level than \(\kappa_\mu=(\mu/\ell)^{1/2}\), where \(\ell\) satisfies th equality \(\lambda=(\mu/\ell)^{(p-2)/p}\). Such critical points of \(E(\cdot,{\mathcal{G}})\) solve a nonlinear Schrödinger equation on every edge of \({\mathcal{E}}\).