The duality method for mean field games systems

Abstract

<p style='text-indent:20px;'>In this paper we prove, using a duality method, existence of solutions for the nonlinear elliptic mean field games type system</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \left\{ \begin{array}{cl} -{\mathop{{{\rm{div}}}}}(M(x)\,{\nabla} u) + u - {\mathop{{{\rm{div}}}}}(u\,A(x)\,{\nabla}\psi) = f(x) &amp; {\rm{in \; \Omega ,}}\\ -{\mathop{{{\rm{div}}}}}(M(x)\,{\nabla}\psi) + \psi + A(x)\,{\nabla}\psi \cdot {\nabla} \psi = u^{p-1} &amp; {\rm{in \; \Omega ,}} \\ u = 0 = \psi &amp; {\rm{on \; \partial\Omega ,}} \end{array} \right. $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>under different assumptions on <inline-formula><tex-math id="M1">\begin{document}$ p &gt; 1 $\end{document}</tex-math></inline-formula>, and the function <inline-formula><tex-math id="M2">\begin{document}$ f(x) $\end{document}</tex-math></inline-formula> in Lebesgue spaces.</p>