2013/12/06 by Desheng Li, Li, Desheng, Guoliang Shi +3
Mathematics · #35B34 #35B40 #35J15 #35K55 #37B25 #Dynamical Systems (math.DS) #FOS: Mathematics #math.DS #msc:35B34 #msc:35B40 #msc:35J15 #msc:35K55 #msc:37B25
paper · pdf · doi:10.48550/arxiv.1312.1868
We have removed the condition of separability on the phase $X$ and some other restrictions (such as nonsingularity of the flow) in this new version
arxiv created 2015/05/18 · arxiv updated 2015/05/19
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 ut-Δu-μu=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 -Δu+a(x)u=f(x,u) on Rn. We prove the existence of positive solutions by applying a mountain pass lemma of semiflows to the parabolic flow of the problem.