This talk deals with elliptic problems of the form (SP)
−\Delta u + V(x)u + K(x)\phi(x)u = u^p,\\
−\Delta\phi = K(x)u^2,\\
u(x) > 0, x \in \mathbb R^3
where $K\ge 0$ and $V \ge V_\infty\ge 0$, $V(x)\to V_\infty$ as $|x|\to \infty$.
The model (SP) describes some physical problems, for example electrostatic situations in which the interaction between an electrostatic field and solitary waves has to be considered. We examine both the subcritical case $p\in(3, 5)$ and the critical case $p = 5$ and analyse the different situations that occur. In particular, in the critical case problem (SP) exhibits a “double” lack of compactness because of the unboundedness of $\mathbb R^3$ and the critical growth of the nonlinear term. Let us remark that ground state solutions of (SP) do not exist in our assumptions. We show some existence results of bound state solutions in both cases. When $K\equiv 0$ and $p = 5$ problem (SP) reduces to a critical Schrödinger equation, for which we get a new result.
These are joint works with Giovanna Cerami.