On a generalized timoshenko-kirchhoff equation

TL;DR

This paper studies a generalized fourth-order nonlinear Kirchhoff equation with sublinear nonlinearity, employing variable transformation and variational methods to establish the existence of solutions in bounded domains.

Contribution

It introduces a reduction technique to transform a complex Kirchhoff problem into a semilinear one and applies variational and topological tools for solution existence proof.

Findings

Existence of solutions for the generalized Kirchhoff equation

Reduction of the problem to a semilinear form

Application of variational and topological methods

Abstract

In this paper we consider a generalized fourth order nonlinear Kirchhoff equation in a bounded domain in $\mathbb R^{N}, N\geq2$ under Navier boundary conditions and with sublinear nonlinearity. We employ a change of variable which reduces the problem to a semilinear one. Then variational and topological tools are used in order to prove the existence of a solution.