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Multiple solutions to relativistic systems with potential boundary conditions
Part of:
Boundary value problems
Ordinary differential operators
Equations and inequalities involving nonlinear operators
Published online by Cambridge University Press: 01 September 2025
Abstract
We consider potential systems of differential equations of the form under the general boundary condition
\begin{equation*}-\left[ \phi(u^{\prime}) \right] ^{\prime} = \nabla_u F(t,u),\quad \mbox{in } [0,T],\end{equation*} where 
\begin{equation*}\left ( \phi \left( u^{\prime }\right)(0), -\phi \left( u^{\prime }\right)(T)\right )\in \partial j(u(0), u(T)),\end{equation*}
$\phi(y)={y}/{\sqrt{1- |y|^2}}$ and 
$j:\mathbb{R}^N \times \mathbb{R}^N \rightarrow (-\infty, +\infty]$ is convex and lower semicontinuous. Making use of the variational approach introduced in the recent paper “Potential systems with singular 
$\phi$-Laplacian”, we obtain multiplicity of solutions when the action functional is even, as well as existence of multiple geometrically distinct solutions when this functional is invariant with respect to some discrete group.
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- © The Author(s), 2025. Published by Cambridge University Press on behalf of The Edinburgh Mathematical Society.
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