Strongly coupled elliptic equations related to mean-field games systems (Q285631)

The authors investigate the problem of a priori estimates in Lebesgue spaces and the existence of solutions to the class of elliptic systems NEWLINENEWLINE\[NEWLINE\begin{cases} {\mathcal L}(\zeta)+\zeta -{\text{div}}(\zeta A(x)\nabla u) = f &\text{ in }\Omega ,\\NEWLINE{\mathcal L}(u)+ u+\theta A(x)\nabla u \cdot \nabla u = \zeta^p &\text{ in }\Omega ,\\NEWLINE\zeta = 0 = u &\text{ on }\partial \Omega, \end{cases} \leqno{(1.3)}NEWLINE\]NEWLINENEWLINEwhere \(p >0\), \(0 < \theta <1\), \(f\geq 0\) is a function in some Lebesgue space \(L^m(\Omega)\), \(m \geq 1\), \(\Omega\) is a bounded, open subset of \(\mathbb R^N\), \(N\geq 2\) and \(M:\Omega \to \mathbb R^{N^2}\), \( A :\Omega \to\mathbb R^{N^2}\) are matrices such thatNEWLINENEWLINE\[NEWLINEM(x)\xi \cdot \xi \geq \alpha |\xi |^2 , \quad |M(x)| \leq \beta , \leqno{(1.1)}NEWLINE\]NEWLINENEWLINEandNEWLINENEWLINE\[NEWLINEA(x)\xi \cdot \xi \geq \alpha |\xi |^2 , \quad |A(x)| \leq \beta , \leqno{(1.2)}NEWLINE\]NEWLINENEWLINEfor every \(\xi\) in \(\mathbb R^N\), where \(0 <\alpha \leq \beta \) are real numbers. Furthermore, \(M\) is symmetric and the differential operator \({\mathcal L} : W^{1,2}_0(\Omega) \to W^{-1,2}(\Omega)\) is defined byNEWLINENEWLINE\[NEWLINE{\mathcal L}(v)=-\text{div}(M(x)\nabla v) ,\quad v \in W^{1,2}_0(\Omega)\,.NEWLINE\]NEWLINENEWLINEThe authors assume \(A(x)\) and \(M(x)\) to be only measurable with respect to \(x\). The main purpose of this paper is to find conditions on the Lebesgue class \(L^m\) of the data \(f\) and the growth exponent \(p\) of the coupling term which allow to find a priori estimates and solutions of the system.NEWLINENEWLINENEWLINENEWLINEWe denote \(T_k(s) = \max(-k, \min(s,k))\), \(k \geq 0\), \(s \in R\). Then we can define what the authors mean by a solution of the system (1.3).NEWLINENEWLINENEWLINENEWLINE{Definition 1.1.} By a ``solution of the system (1.3)'' we mean a couple \((u, \zeta)\) of functions such that \(u\) belongs \, to \(W^{1,2}_0(\Omega)\), \(T_k(\zeta)\) belongs \, to \(W^{1,2}_0(\Omega)\) for every \(k>0\), \(\zeta\) belongs \, to \(L^p(\Omega)\), and \(\zeta\) is an entropy solution of the first equation, in the sense thatNEWLINENEWLINE\[NEWLINE\int_\Omega M(x)\nabla \zeta \cdot \nabla T_k(\zeta -\varphi)+\int_\OmegaNEWLINE\zeta T_k(\zeta - \varphi)+\int_\Omega \zeta A(x)\nabla u \cdot \nabla T_k(\zeta - \varphi) \leq \int_\Omega f T_k(\zeta -\varphi) ,NEWLINE\]NEWLINENEWLINEfor every \(\varphi\) in \(W^{1,2}_0(\Omega) \cap L^\infty (\Omega)\), and for every \(k>0\), while \(u\) is a weak solution of the second one, in the sense thatNEWLINENEWLINE\[NEWLINE\int_\Omega M(x)\nabla u \cdot \nabla\varphi + \int_\Omega u\varphi +\thetaNEWLINE\int_\Omega A(x)\nabla u \cdot \nabla u\varphi = \int_\Omega \zeta^p \varphi,NEWLINE\]NEWLINENEWLINEfor every \(\varphi\) in \(W^{1,2}_0(\Omega) \cap L^\infty (\Omega)\).NEWLINENEWLINENEWLINENEWLINEThe following theorems are the main results of this paper.NEWLINENEWLINENEWLINENEWLINE{Theorem 1.2.} Let \(p>0\) and let \(f\geq 0\) be such thatNEWLINENEWLINE\[NEWLINE\begin{cases}NEWLINEf \text{ belongs to } L^1(\Omega) &\text{ if } 0<p<\frac{2}{N-2},\\NEWLINEf \text{ belongs to } L^1 \log L^1(\Omega) &\text{ if } p = \frac{2}{N-2} ,\\NEWLINEf \text{ belongs to } L^m(\Omega), m = \frac{2N(p+1)}{(N+2)(p+1)+N} &\text{ if }p >\frac{2}{N-2}.\end{cases}NEWLINE\]NEWLINENEWLINEThen there exists a solution \((u, \zeta)\) of the system (1.3), in the sense of Definition 1.1, with \(u\) in \(W^{1,2}_0(\Omega)\), and \(\zeta\) in \(L^{p+1}(\Omega)\). Furthermore, NEWLINENEWLINE\[NEWLINE \begin{cases} u \text{ belongs to } L^\infty(\Omega)&\text{ if } 0<p< \frac{2}{N-2},\\NEWLINEu \text{ belongs to } L^s, \text{ for every } s\geq 1 &\text{ if } p = \frac{2}{N-2},\\NEWLINEu \text{ belongs to } L^Q(\Omega),\;Q = \frac{2N(p+1)}{(N-2)p-2}&\text{ if } p > \frac{2}{N-2},\end{cases} \leqno{(1.5)}NEWLINE\]NEWLINENEWLINEand \(\zeta|\nabla u|^2\) belongs \, to \(L^1(\Omega)\). Finally, if \(f\) belongs \, to \(L^m(\Omega)\) withNEWLINENEWLINE\[NEWLINE\begin{cases} NEWLINEm>1 &\text{ if } 0<p \leq \frac{2}{N-2},\\NEWLINEm \geq \frac{2N(p+1)}{(N+2)(p+1)+N} &\text{ if } p \geq \frac{2}{N-2},\end{cases}NEWLINE\]NEWLINENEWLINEthen \(\zeta\) belongs to \(W^{1,q}_0(\Omega)\), \(q=\frac{2(p+1)}{p+2}\).NEWLINENEWLINENEWLINENEWLINEThe second theorem deals with the remaining cases, also giving some summability results on \(u\) and \(\zeta\).NEWLINENEWLINENEWLINENEWLINE{Theorem 1.3.} Let \(p>\frac{2}{N-2}\) and let \(f\) in \(L^m(\Omega)\) withNEWLINENEWLINE\[NEWLINE\begin{cases}NEWLINEm \geq 1 &\text{ if }\frac{2}{N-2} <p<2^*,\\NEWLINEm>\frac{2Np}{(N+2)p+2N} &\text{ if } p \geq 2^*.NEWLINE\end{cases}NEWLINE\]NEWLINENEWLINEThen there exists a solution \((u, \zeta)\) of system (1.3), in the sense of Definition 1.1. Furthermore,NEWLINENEWLINE\[NEWLINE\begin{cases}NEWLINEu \text{ belongs to } L^q(\Omega), \text{ for every } 1 \leq q < \frac{N(p+2)}{Np-2(p+1)}, &\text{ if } m = 1,\\NEWLINEu \text{ belongs to } L^Q(\Omega), \text{ with } Q= \frac{Nm(p+2)}{Np-2m(p+1)},&\text{ if } \frac{2Np}{(N+2)p+2N} <m<\frac{N}{2} \frac{p}{p+1},\\NEWLINEu \text{ belongs to } L^q(\Omega), \text{ for every } q \geq 1, &\text{ if } m \geq \frac{N}{2} \frac{p}{p+1},NEWLINE\end{cases}NEWLINE\]NEWLINENEWLINEandNEWLINENEWLINE\[NEWLINE\begin{cases} NEWLINE\zeta \text{ belongs to } L^s(\Omega), \text{ for every } 1 \leq s < \frac{p+2}{2} \frac{N}{N-1}, &\text{ if } m = 1,\\NEWLINE\zeta \text{ belongs to } L^s(\Omega), \text{ with } s = \min(p + 1,\frac{p+2}{2} m^*), &\text{ if } m>\frac{2Np}{(N+2)p+2N}.NEWLINE\end{cases}NEWLINE\]