Unilateral global interval bifurcation and one-sign solutions for Kirchhoff type problems

Abstract

In this paper, we study the following Kirchhoff type problems: $ \left\{ \begin{array}{l} -(\int_{\Omega}|\nabla u|^{2}dx)\Delta u = \lambda u^{3}+g(u, \lambda), \, \, \, \, \, \, \, \, \mathrm{in}\, \, \Omega,\\ u = 0, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \mathrm{on}\, \, \partial\Omega, \end{array} \right. $ where $ \lambda $ is a parameter. Under some natural hypotheses on $ g $ and $ \Omega $, we establish a unilateral global bifurcation result from interval for the above problem. By applying the above result, under some suitable assumptions on nonlinearity, we shall investigate the existence of one-sign solutions for a class of Kirchhoff type problems.

Keywords

• unilateral global interval bifurcation
• one-sign solutions
• kirchhoff type problems