  In this paper we prove some linking theorems and mountain pass type results
for dynamical systems in terms of local semiflows on complete metric spaces.
Our results provide an alternative approach to detect the existence of compact
invariant sets without using the Conley index theory. They can also be applied
to variational problems of elliptic equations without verifying the classical
P.S. Condition.
  As an example, we study the resonant problem of the nonautonomous parabolic
equation $ u_t-\Delta u-\mu u=f(u)+g(x,t) $ on a bounded domain. The existence
of a recurrent solution is proved under some Landesman-Laser type conditions by
using an appropriate linking theorem of semiflows. Another example is the
elliptic equation $-\Delta u+a(x)u=f(x,u)$ on $R^n$. We prove the existence of
positive solutions by applying a mountain pass lemma of semiflows to the
parabolic flow of the problem.
