The Nehari Manifold for p-Laplacian Equation with Dirichlet Boundary Condition

Abstract

The Nehari manifold for the equation −∆pu(x) = λu(x)|u(x)| p−2 + b(x)|u(x)| γ−2u(x) for x ∈ Ω together with Dirichlet boundary condition is investigated in the case where 0 < γ < p. Exploiting the relationship between the Nehari manifold and fibrering maps (i.e., maps of the form of t → J(tu) where J is the Euler functional associated with the equation), we discuss how the Nehari manifold changes as λ changes, and show how existence results for positive solutions of the equation are linked to the properties of Nehari manifold.

Keywords

• the p-Laplacian
• variational methods
• Nehari manifold
• fibrering maps