Positive solutions of an obstacle problem
In this paper the author studies the existence of positive solutions of the obstacle problem, with a nonlinear term in the variational inequality. The variational problem is formulated in an exterior domain with a smooth boundary. A positive parameter \(\lambda\) is included in the variational inequality and a discussion is done according to the values of this parameter. It is proved the existence of a value \(0<\lambda^*<+\infty\) such that the problem has at least one positive solution for every \(\lambda\) in \((0,\lambda^*)\) and there is no positive solution for \(\lambda>\lambda^*\). Furthermore, there exists a second value \(\lambda^{**}\), with \(0<\lambda^{**}\leq \lambda^*\), such that the problem has at least two different positive solutions for every \(\lambda\in (0,\lambda^{**})\). Finally, assuming that the obstacle belongs to \(H^{2,q}(\Omega)\), the author proves that any positive solution belongs to the space \(H^{2,q}(\Omega)\cap C^{1,\alpha}(\Omega)\).
scientific article; zbMATH DE number 824220
scientific article