On existence of a unique generalized solution to systems of elliptic PDEs at resonance (Q2872671)

Let \(\Omega\) be a bounded domain in \(\mathbb{R}^n\) with \(C^2\)-boundary and let \(L_k(\varphi)=\sum_{i=1}^n\sum_{i=1}^n\frac{\partial}{\partial x_i}\left(a_{ij}^{(k)}(x)\frac{\partial \varphi}{\partial x_j}\right)\), \(k=1,\dots,m\), be \(m\) symmetric strongly elliptic second-order differential operators, where the functions \(a_{ij}^{(k)}\) are of class \(C^1\) in \(\overline{\Omega}\). Moreover, let \(g:\Omega \times \mathbb{R}^m\rightarrow \mathbb{R}^m\) be a continuous function which is continuously differentiable in the second variable, and let \(h\in L^2(\Omega, \mathbb{R}^m)\).NEWLINENEWLINE The authors consider the following boundary value problem: \(Lu+g(x,u)=h(x)\) in \(\Omega\), \(u=0\) on \(\partial \Omega\), where \(Lu=(L_1u_1,\dots,L_m u_m)\) for \(u=(u_1,\dots,u_m)\), and prove the existence of a unique generalized solution by assuming the following further assumption on the function \(g\):NEWLINENEWLINE\noindent 1) \(\frac{\partial g(x,u)}{\partial u}\) is symmetric;NEWLINENEWLINE\noindent 2) there exist two continuous functions \(\alpha,\beta:[0,\infty)\rightarrow (0,\infty)\) with \(\int_1^\infty\min\{\alpha(s),\beta(s)\}ds=\infty\) and two constant symmetric \(m\times m\) matrices \(B_1\) and \(B_2\) with eigenvalues \(\lambda_{N_1}^{1},\dots,\lambda_{N_m}^{m}\) and \(\lambda_{N_{2}}^{1},\dots,\lambda_{N_{m+1}}^{m}\), respectively, \(\lambda_{N_k}^k\) and \(\lambda_{N_{k+1}}^k\) being two consecutive eigenvalues of \(-L_k\varphi=\lambda \varphi\) in \(\Omega\), \(\varphi=0\) on \(\Omega\), such that \(B_1+\alpha(\|u\|)I\leq \frac{\partial g(x,u)}{\partial u}\leq B_2-\beta(\|u\|)I\) on \(\mathbb{R}^m\times \mathbb{R}^m\).NEWLINENEWLINE\noindent This result extends to more general differential operators a similar result for the Laplacian operator established in [\textit{T. Qiao} and \textit{W. Li}, J. Nanjing Univ., Math. Biq. 24, No. 1, 29--34 (2007; Zbl 1174.35034)].