  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 $\mathbb{R}^N$ \[ \mathcal{P}[u] = f(x,u,\nabla u), \quad x\in
\mathbb{R}^N, \] 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.
