2016/08/21 by Ricceri, Biagio
#Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Functional Analysis (math.FA)
paper · doi:10.48550/arxiv.1608.05903
Let BL the open ball in \bf Rn centered at 0, of radius L, and let ϕ be a homeomorphism from BL onto \bf Rn such that ϕ(0)=0 and ϕ=∇Φ, where the function Φ: BL→ ]-∞,0] is continuous and strictly convex in BL, and of class C1 in BL. Moreover, let F:[0,T]× \bf Rn→ \bf R be a function which is measurable in [0,T], of class C1 in \bf Rn and such that ∇xF satisfies the L1-Carathéodory conditions. Set K=\u∈ Lip([0,T],\bf Rn) : |u'(t)|≤ L for a.e. t∈ [0,T] , u(0)=u(T)\ , and define the functional I:K→ \bf R by I(u)=∫0T(Φ(u'(t))+F(t,u(t)))dt for all u∈ K. In [1], Brezis and Mawhin proved that any global minimum of I in K is a solution of the problem \cases(ϕ(u'))'=∇xF(t,u) amp; in [0,T]\cr amp; \cr u(0)=u(T) , u'(0)=u'(T) .\cr In the present paper, we provide a set of conditions under which the functional I has at least two global minima in K. This seems to be the first result of this kind. The main tool of our proof is the well-posedness result obtained in [3].