Multiple Periodic Solutions of Lagrangian Systems of Relativistic Oscillators

Let B L the open ball in R n centered at 0, of radius L, and let ϕ be a homeomorphism from B L onto R n such that ϕ(0) = 0 and ϕ = ∇Φ, where the function Φ : B L ¯ → ] − ∞ , 0 ] $$\varPhi :\overline {B_L}\to ]-\infty ,0]$$ is continuous and strictly convex in B L ¯ $$\overline {B_L}$$ , and of class C 1 in B L . Moreover, let F : [0, T] ×R n →R be a function which is measurable in [0, T], of class C 1 in R n and such that ∇ x F satisfies the L 1-Carathéodory conditions. Set K = { u ∈ Lip ( [ 0 , T ] , R n ) : | u ′ ( t ) | ≤ L f o r a . e . t ∈ [ 0 , T ] , u ( 0 ) = u ( T ) } , $$\displaystyle K=\{u\in \mathrm {Lip}([0,T],{\mathbf {R}}^n) : |u'(t)|\leq L\hskip 5pt for\hskip 3pt a.e.\hskip 3pt t\in [0,T] , u(0)=u(T)\}\ , $$ and define the functional I : K →R by I ( u ) = ∫ 0 T ( Φ ( u ′ ( t ) ) + F ( t , u ( t ) ) ) d t $$\displaystyle I(u)=\int _0^T(\varPhi (u'(t))+F(t,u(t)))dt $$ for all u ∈ K. In Brezis and Mawhin (Commun. Appl. Anal. 15:235–250, 2011), proved that any global minimum of I in K is a solution of the problem ( ϕ ( u ′ ) ) ′ = ∇ x F ( t , u ) in [ 0 , T ] u ( 0 ) = u ( T ) , u ′ ( 0 ) = u ′ ( T ) . $$\displaystyle \left \{ \begin {array}{ll} (\phi (u'))'=\nabla _xF(t,u) & \mbox{ in } \mathrm {[0,T]}\\ \ & \ \\ u(0)=u(T)\ , \hskip 3pt u'(0)=u'(T)\ . \end {array} \right . $$ 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 B. Ricceri (J. Global Optim. 40:389–397, 2008).