In this paper we study the Boundary value problem [ left{ begin{array}{ll} -Delta u+ eps qPhi f(u)=eta|u|^{p-1}u & ext{in } Omega, \ - Delta Phi=2 qF(u)& ext{in } Omega, \ u=Phi=0 & ext{on }partial Omega, end{array} ight.] where $Omega subset mathbb{R}^3$ is a smooth bounded domain, $1 < p < 5$, $eps ,eta= pm 1$, $q>0$, $f:R oR$ is a continuous function and $F$ is the primitive of $f$ such that $F(0)=0.$ We provide existence and multiplicity results assuming on $f$ a subcritical growth condition. The critical case is also considered and existence and nonexistence results are proved.

This paper deals with the system [ left{ begin{array}{ll} -Delta u = lambda u + q |u|^3 u phi & hbox{in } B_R,\ -Delta phi = q |u|^5 & hbox{in } B_R,\ u = phi = 0 & hbox{on } partial B_R. end{array} ight. ] We prove existence and nonexistence results depending on the value of $lambda$.

In this paper we prove the existence of a nontrivial non-negative radial solution for a quasilinear elliptic problem. Our aim is to approach the problem variationally by using the tools of critical points theory in an Orlicz-Sobolev space. A multiplicity result is also given.

In this paper we study semiclassical states for the problem [ -varepsilon^2 Delta u + V(x) u = f(u) qquad hbox{in} mathbb{R}^N, ] where $f(u)$ is a superlinear nonlinear term. Under our hypotheses on $f$ a Lyapunov-Schmidt reduction is not possible. We use variational methods to prove the existence of spikes around saddle points of the potential $V(x)$.