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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

Pasquale Imparato
Roma, Italy

Luigi Orsina
UniversitÓ La Sapienza, Roma, Italy


\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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