A local minimum theorem and critical nonlinearities

Abstract

In this paper the existence of two positive solutions for a Dirichlet problem having a critical growth, and depending on a real parameter, is established. The approach is based on methods which are totally variational, unlike the fundamental result of Ambrosetti, Brezis and Cerami where a clever combination of topological and variational methods is used in order to obtain the same conclusion. In addition, a numerical estimate of real parameters, for which the two solutions are obtained, is provided. Our main tool is a local minimum theorem.

Keywords

• critical growth
• nonlinear differential problem
• variational methods
• palais- smale condition
• local minimum
• 35j60
• 35j20
• 58e30
• 49j52
• 49j50