| Contents |
| We want to discuss the use of multiple constraints in order to describe the set of the non-negative solutions of the model system
\[
\left\{\begin{aligned}
-\Delta u_1(x) +\omega_1 u_1(x) & = \left( \mu_1 u_1^2 (x) +\beta u_2^2(x) \right) u_1(x) \\
-\Delta u_2(x) +\omega_2 u_2(x) & = \left( \mu_2 u_2^2 (x) +\beta u_1^2(x) \right) u_2(x) \\
\end{aligned}\right. \qquad x\in\mathbb{R}^n
\]
where \(n=1,2,3\) and \(\beta,\mu_i,\omega_i\) (for \(i=1,2\)) are real positive numbers.\\
Existence and multiplicity of non-negative and positive solutions will be discussed. In particular we will try to characterize the set of the parameters for which solutions, with both positive components, exist. \\
Moreover we will present also some generalizations and open problems. |
|