Constrained Semilinear Elliptic Systems on ${\mathbb{R}}^N$

We prove the existence of solutions $u$ in $H^1({\mathbb{R}}^N,{\mathbb{R}}^M)$ of the following strongly coupled semilinear system of second order elliptic PDEs on ${\mathbb{R}}^N$ $$\mathcal{P}[u]=f(x,u,\mathrm{\nabla }u),x\in {\mathbb{R}}^N,$$ whith pointwise constraints. We present the construction of the suitable topoligical degree which allows us to solve the above system on bounded domains. The key step in the proof consists of showing that the sequence of solutions of the truncated system is compact in $H^1$ by the use of the so-called tail estimates.