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April, 2021 Existence of Two Periodic Solutions to General Anisotropic Euler-Lagrange Equations
Magdalena Chmara
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Taiwanese J. Math. 25(2): 409-425 (April, 2021). DOI: 10.11650/tjm/200902


This paper is concerned with the following Euler-Lagrange system \[ \begin{cases} \frac{d}{dt} \mathcal{L}_v(t,u(t),\dot{u}(t)) = \mathcal{L}_x(t,u(t),\dot{u}(t)) \quad \textrm{for a.e. $t \in [-T,T]$}, \\ u(-T) = u(T), \\ \mathcal{L}_v(-T,u(-T),\dot{u}(-T)) = \mathcal{L}_v(T,u(T),\dot{u}(T)), \end{cases} \] where Lagrangian is given by $\mathcal{L} = F(t,x,v) + V(t,x) + \langle f(t), x \rangle$, 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.


I would like to thank the referee for valuable comments and suggestions.


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Magdalena Chmara. "Existence of Two Periodic Solutions to General Anisotropic Euler-Lagrange Equations." Taiwanese J. Math. 25 (2) 409 - 425, April, 2021.


Received: 12 March 2020; Revised: 9 July 2020; Accepted: 9 September 2020; Published: April, 2021
First available in Project Euclid: 24 March 2021

Digital Object Identifier: 10.11650/tjm/200902

Primary: 46E30 , 46E40

Keywords: anisotropic Orlicz-Sobolev space , Euler-Lagrange equations , Mountain pass theorem , Palais-Smale condition

Rights: Copyright © 2021 The Mathematical Society of the Republic of China


Vol.25 • No. 2 • April, 2021
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