Variational approach for a class of cooperative systems
The main result of the paper establishes the existence and uniqueness of positive minimizers \(u\in H^1_0(\Omega)\) of the Euler-Lagrange equation \(D_u\mathcal E(u)v=0\), with NEWLINE\[NEWLINE\mathcal E(u)=\int_\Omega \left( |\nabla u|^2-\lambda u^2-\gamma|(-\Delta-\lambda)^{-1/2}u|^2\right)/2\mathrm{dx} +\int_\Omega a\int_0^u f(\cdot,\xi)\xi d\xi\mathrm{dx}.NEWLINE\]NEWLINE The result is proved through classical variational methods under the assumptions: \(\gamma>0\); the domain \(\Omega\subset\mathbb R^n\) (\(n\geq 1\)) is bounded and of class \(C^{2+\mu}\) for some \(\mu\in ]0,1[\); the function \(a\in C^{\mu}(\bar\Omega)\) is such that \(a^{-1}(\{0\})=\bar\Omega\setminus\Omega_+\) and \(\bar\Omega_+\subset\Omega\), where \(\Omega_+:=\{x\in\Omega:\;a(x)>0\}\) is of class \(C^{2+\mu}\); the function \(f\in C^{\mu, 1+\mu}(\bar\Omega \times [0,\infty[)\) obeys some conditions that imply that \(f(\cdot,u)\rightarrow \infty\) as \(u\rightarrow \infty\) uniformly in \(\bar\Omega\); and \(\lambda\in\mathbb R\) is strictly lesser than the principal eigenvalue of the Laplace operator in \(\Omega\) under homogeneous Dirichlet boundary conditions. The limiting behavior of the positive minimizers is also analyzed as the parameter \(\gamma\) approaches the limiting values for which the existence of positive solutions is held. Furthermore, the map \(\gamma\mapsto u\) is shown that it is increasing and of class \(C^1\).
scientific article; zbMATH DE number 6082080
scientific article