Regularizing effect for a system of Schrödinger-Maxwell equations
Let \(\Omega\) be an open bounded subset of \(\mathbb{R}^N\) (\(N > 2\)), and let \(r>1\), \(A>0\), and \(f\in L^m(\Omega)\), with \(1< m<\frac{2N}{N+2}\). The authors study the following Dirichlet problem for a system of Schrödinger-Maxwell equations \(-\)div\((M(x)|\nabla u|)+A\varphi|u|^{r-2}u=f\), \ \ \ in \(\Omega\), \(-\)div\((M(x)|\nabla \varphi|)=|u|^r\), \ \ \ in \(\Omega\), \(u=\varphi=0\), \ \ \ on \(\partial \Omega\), where \(M(x)\) is a symmetric measurable matrix satisfying, for some \(\alpha,\beta>0\), \((M(x)\xi)\xi\geq \alpha|\xi|^2\) \ and \ \(|M(x)|\leq \beta\), for a.e. \(x\in\Omega\) and for all \(\xi \in \mathbb{R}^N\). The main result of this paper states that, for \(r>\frac{2N}{N-2}\) and \(m>\frac{r}{r-1}\), there exists a solution of finite energy \((u,\varphi)\in W_0^{1,2}(\Omega)\times W_0^{1,2}(\Omega)\) to the problem above. In the case \(1<r<\frac{2n}{N-2}\), the same result ensures the existence of a solution \((u,\varphi)\in W_0^{1,\frac{m}{m-1}}(\Omega)\times W_0^{1,q}(\Omega)\), provided that \(m>\max\{1,Nr/(N+2r)\}\). Here, \(q=2\) if \(m\geq 2Nr/(N+2+4r)\), and \(q=Nm/(Nr-2mr-m)\) if \(m<\max{1,Nr/(N+2r)}\). The authors emphasize the fact that when \(r>\frac{N+2}{N-2}\), then \(|u|^r\notin L^{\frac{2N}{N+2}}(\Omega)\) and , in consequence, the condition \(\varphi \in W_0^{1,2}(\Omega)\) is not deducible, in this case, from the fact that \(\varphi\) is a solution of the second equation. Actually, this condition is a consequence of the regularizing effect on \(\varphi\) that derives from being \((u,\varphi)\) a solution of the system above. To proof the main result, the authors first show that, for \(f\in L^\infty(\Omega)\), the system admits a solution \((u,\varphi)\in W_0^{1,2}(\Omega)\times W_0^{1,2}(\Omega)\), with \(u,\varphi\in L^\infty(\Omega)\), for all \(r>1\). This solution is a saddle point of the energy functional associated to the system. Then, using a truncation argument and gradient estimates, the authors find sequences of solutions \(\{u_n\},\{\varphi_n\}\) of approximating equations, with bounded right-hand side \(f_n\), such that \((u_n,\varphi_n)\) converges to a solution of the system.
scientific article; zbMATH DE number 6824190