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Minimax Theory and its Applications 08 (2023), No. 2, 319--332
Copyright Heldermann Verlag 2023



Saddle Points of some Integral Functionals and Solutions of Elliptic Systems

Lucio Boccardo
Istituto Lombardo, Milano, Italy
boccardo@mat.uniroma1.it

Pasquale Imparato
Roma, Italy
pasquale.imparato1994@gmail.com

Luigi Orsina
UniversitÓ La Sapienza, Roma, Italy
orsina@mat.uniroma1.it



[Abstract-pdf]

\def\w{W_{0}^{1,2}(\Omega)} \def\wp{W_{0}^{1,p}(\Omega)} We prove the existence of finite energy solutions $u$ and $\psi$ for two systems, one of which is \begin{equation*} \left\{\ \begin{aligned} & u \in \w:\hskip8pt -{\rm div}\,(a(x)\,\nabla u) = -{\rm div}\,(\psi\,E(x))\,, \\[1mm] & \psi \in \wp:\ \ -{\rm div}\,(a(x)\,|\nabla\psi|^{p-2}\,\nabla\psi) + E(x) \cdot \nabla u = f(x)\,, \end{aligned} \right. \end{equation*} under some assumptions on $p$ and on the vector field $E(x)$.

Keywords: Integral functions, saddle points, nonlinear elliptic equations.

MSC: 35J20, 35J47, 35J62.

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