| Contents |
| We consider the existence of solutions in $H^1(R^N) \times H^1(R^N)$ for systems of the form
\begin{equation}
\left\{
\begin{array}{ll}
- \Delta u_1 - \lambda_1 u_1 = \mu_1 |u_1|^{p_1 -2}u_1 + r_1 \beta |u_1|^{r_1-2}|u_2|^{r_2}u_1 \\
- \Delta u_2 - \lambda_2 u_2 = \mu_2 |u_2|^{p_2 -2}u_2 + r_2 \beta |u_1|^{r_1}|u_2|^{r_2 -2}u_2.
\end{array}
\right.
\end{equation}
Here $N \geq 1,$ $ \mu_1, \mu_2, r_1, r_2$ are given positive constants and $\beta \in R$. For $a_1 >0, a_2 >0$ given, we look for solutions satisfying $||u_1||_2^2 = a_1$ and $||u_2||_2^2 = a_2$. In the system $\lambda_1$ and $\lambda_2$ are unknown and they will appear as Lagrange parameters. This talk describes joint work with T. Bartsch |
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