Existence of two periodic solutions to general anisotropic Euler-Lagrange equations

2019/10/31 by Magdalena Chmara, Chmara, M.
Mathematics · #34C25 #37J45 #46E40 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Differential Equations Analysis #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.1911.00150

openalex publication_date 2019/10/31 · openalex created_date 2019/11/22 · openalex updated_date 2026/07/28

Abstract

This paper is concerned with the following Euler-Lagrange system (d)/(dt)L_v(t,u(t), u(t))=L_x(t,u(t), u(t)) for a.e. t∈[-T,T], u(-T)=u(T), where Lagrangian is given by L=F(t,x,v)+V(t,x)+⟨ f(t), x⟩, growth conditions are determined by an anisotropic G-function and some geometric conditions at infinity. We consider two cases: with and without forcing term f. Using a general version of the Mountain Pass Theorem and Ekeland's variational principle we prove the existence of at least two nontrivial periodic solutions in an anisotropic Orlicz-Sobolev space.

Citations

• A relation between pointwise convergence of functions and convergence of functionals

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