Some properties for the first eigenvalue of nonlinear weighted problems and applications

We prove some properties of the first eigenvalue of the problem \begin{array}{ll} -{\cal A}_p u \colon = - \hbox{\rm div\ } \Big( (A\D u, \D u)^{(p-2)/2}A\D u\Big)= \lambda V(x) |u|^{p-2} u & \hbox{\rm in\ } \O \\ \quad u=0 & \hbox{\rm on\ } \partial \O . \end{array} In particular, the first eigenvalue is shown to be isolated. Moreover, existence and non existence results of solutions in W^{1, p}_0(\Omega) for nonlinear weighted equations with exponential growth are obtained.