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

Abstract

<abstract><p>In this paper, we study the following Kirchhoff type problems:</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \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. $\end{document} </tex-math></disp-formula></p> <p>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.</p></abstract>