Existence of Solutions to a Class of Kirchhoff-Type Equation with a General Subcritical Nonlinearity

Abstract

This paper is devoted to the following nonlinear Kirchhoff-type problem $$-\bigg(a + b\int_{\Omega}|\nabla u|^{2}\,{d}x\bigg)\Delta u = \lambda f(x,u)$$ with the Dirichlet boundary value. We show that the Kirchhoff-type problem has at least a weak nontrivial solution for all \({\lambda > 0}\) under suitable assumptions on the nonlinear term f with more general growth condition. The main tools are variational method, critical point theory and some analysis techniques.