Constrained semilinear elliptic systems on $\mathbb R^N$

The existence of solutions $u$ in $H^1(\mathbb R^N,\mathbb R^M)\cap H^2_{loc}(\mathbb R^M)$ a coupled semilinear system second order elliptic partial differential equations on $\mathbb R^N$ the form \[ \mathcal{P}[u] = f(x,u,\partial u), \quad x\in \mathbb R^N, \] under pointwise constraints is considered. problem studied via constructed suitable topological invariant, so-called constrained degree, which allows to get abstract problems considered as $L^2$-realizations approximating sequence systems obtained by truncation initial bounded subdomains. key step proof consists showing relative $H^1$-compactness truncated use tail estimates. constructions rely semigroup approach combined with methods, well invariance/viability techniques.