$SU(2)$ Yang-Mills-Higgs functional with Higgs self-interaction on $3$-manifolds

Fixing a constant $\lambda >0$, for any parameter $\epsilon >0$ we study critical points of the Yang–Mills–Higgs energy $${\mathcal{Y}}_{\epsilon }(\mathrm{\nabla },\mathrm{\Phi })={\int }_M{\epsilon }^2\mathrm{\mid }{F}_{\mathrm{\nabla }}{\mathrm{\mid }}^2+\mathrm{\mid }\mathrm{\nabla }\mathrm{\Phi }{\mathrm{\mid }}^2+\frac{\lambda }{4{\epsilon }^2}(1-\mathrm{\mid }\mathrm{\Phi }{\mathrm{\mid }}^2{)}^2,$$ defined for pairs $(\mathrm{\nabla },\mathrm{\Phi })$, where $\mathrm{\nabla }$ is a connection on an $SU(2)$-bundle over an oriented Riemannian $3$-manifold $(M^3,g)$, and $\mathrm{\Phi }$ a section of the associated adjoint bundle. When $M$ is closed, we use a $2$-parameter min-max construction to produce, for $\epsilon {\ll }_{M}1$, non-trivial critical points in the energy regime $$1{\lesssim }_{\lambda }{\epsilon }^{-1}{\mathcal{Y}}_{\epsilon }({\mathrm{\nabla }}_{\epsilon },{\mathrm{\Phi }}_{\epsilon }){\lesssim }_{\lambda ,M}1.$$ When $b_1(M)=0$, these critical points are irreducible: ${\mathrm{\nabla }}_{\epsilon }{\mathrm{\Phi }}_{\epsilon }\mathrm{\ne }0$. Next, assuming $M$ has bounded geometry (not necessarily compact), and given critical points with ${\epsilon }^{-1}{\mathcal{Y}}_{\epsilon }({\mathrm{\nabla }}_{\epsilon },{\mathrm{\Phi }}_{\epsilon })$ uniformly bounded, we show that as $\epsilon \to 0$, the energy measures ${\epsilon }^{-1}{e}_{\epsilon }({\mathrm{\nabla }}_{\epsilon },{\mathrm{\Phi }}_{\epsilon })vo{l}_g$ converge subsequentially to $$\mathrm{\mid }h{\mathrm{\mid }}^{2}vo{l}_g+\sum _{x\in S}\mathrm{\Theta }(x){\delta }_x,$$ where $h$ is an $L^2$ harmonic $1$-form, $S$ a finite set and each $\mathrm{\Theta }(x)$ equals the energy of a finite collection of ${\mathcal{Y}}_1$-critical points on ${\mathbb{R}}^3$. Finally, the estimates involved also lead to an energy gap for critical points on $3$-manifolds with bounded geometry. As a byproduct of our results, we deduce the existence of non-trivial ${\mathcal{Y}}_1$-critical points over ${\mathbb{R}}^3$ for any $\lambda >0$.