The Nehari manifold for a semilinear elliptic equation with a sign-changing weight function. (English) Zbl 1074.35032

Summary: The Nehari manifold for the equation \[ -{\Delta}u(x) = {\lambda}a(x)u(x)+b(x)| u(x)|^{{\nu}-1}u(x) \] for \(x{\in}{\Omega}\) together with Dirichlet boundary conditions is investigated. Exploiting the relationship between the Nehari manifold and fibrering maps (i.e., maps of the form \(t{\to}J(tu)\) where \(J\) is the Euler functional associated with the equation) we discuss how the Nehari manifold changes as \({\lambda}\) changes and show how existence and non-existence results for positive solutions of the equation are linked to properties of the manifold.


35J20 Variational methods for second-order elliptic equations
35J60 Nonlinear elliptic equations
47J30 Variational methods involving nonlinear operators
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
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