Existence and bifurcation behavior of positive solutions for a class of Kirchhoff-type problems

Abstract

Abstract We study the existence of positive solutions for the nonlinear Kirchhoff-type equation − a + b ∫ Ω | ∇ u | 2 d x Δ u = α u + β u 3 with Dirichlet boundary condition, where α , β ∈ R are two real parameters. We provide a description of a two-dimensional set in the ( α , β ) plane, which correspond to the existence of positive solutions for the above Kirchhoff type equation.