Existence of Nontrivial Solutions of p-Laplacian Equation with Sign-Changing Weight Functions. (12th February 2014) - Abdeljabbar, Ghanmi
Abstract : This paper shows the existence and multiplicity of nontrivial solutions of the p -Laplacian problem - Δ p u = 1 / σ ( ∂ F ( x, u ) / ∂ u ) + λ a ( x ) | u | q - 2 u for x ∈ Ω with zero Dirichlet boundary conditions, where Ω is a bounded open set in ℝ n, 1 < q < p < σ < p * ( p * = n p / ( n - p ) if p < n, p * = ∞ if p ≥ n ), λ ∈ ℝ ∖ { 0 }, a is a smooth function which may change sign in Ω ̅, , and F ∈ C 1 ( Ω ̅  ×  ℝ, ℝ ) . The method is based on Nehari results on three submanifolds of the space W 0 1, p ( Ω ) .