Existence results for some functional elliptic equations.
Let \(\Omega \) be a bounded open subset of \(\mathbb {R}^{d}\) and let \(A: \Omega \times L^{p}(\Omega)\rightarrow \mathbb {R}\), \(p\geq 1\), such that \(x\in \Omega \mapsto A(x,u)\), be measurable for all \(u\in L^{p}(\Omega)\) and \(u\in L^{p}(\Omega)\mapsto A(x,u)\) be continuous for a.e. \(x\in \Omega \). Moreover, let \(a_{0}\leq A(x,u)\leq a_{\infty}\) for a.e. \(x\in \Omega \) and for every \(u\in L^{p}(\Omega)\), where \(a_{0},\, a_{\infty}>0\). Finally, let \(f: [0,+\infty [\,\rightarrow [0,+\infty [\) be a Lipschitz continuous function such that \(f(\theta_{1})=f(\theta_{2})=0\) and \(f(\theta)>0\) for \(\theta \in \,]\theta_{1},\theta_{2}[\,\), where \(0\leq \theta_{1}<\theta_{2}\). Theorem 1. For \(\lambda >0\) sufficiently large there exists a weak solution \(u_{\lambda}\in H_{0}^{1}(\Omega)\) of \(-A(\cdot ,u_{\lambda})\Delta u_{\lambda}=\lambda f\circ u_{\lambda}\) such that \(0<u_{\lambda}(x)\leq \theta_{2}\) for a.e. \(x\in \Omega \), \(| u_{\lambda}|_{\infty}>\theta_1\); if \(f\) has \(n\) loops the problem has at least \(n\) distinct solutions. Theorem 2. Let \[ J_{\lambda}(u)=\frac {1}{2}\int_{\Omega}| \nabla u| ^{2}\,dx-\lambda \int_{\Omega}^{u(x)} {1}_{]\theta_{1},\theta_{2}[}f\, ds\, dx \] for every \(u\in H_{0}^{1}(\Omega)\) and \(\psi_{\lambda}\) a minimizer of \(J_{\lambda}\). Then, \(\psi_{\lambda}\rightarrow \theta_{2}\) and \(u_{\lambda}\rightarrow \theta_{2}\) in \(L^{p}(\Omega)\) as \(\lambda \rightarrow \infty \).
scientific article; zbMATH DE number 6424467