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

In this paper, we study the following Kirchhoff type problems:

$$\{\begin{array}{l}-({\int }_{\mathrm{\Omega }}|\mathrm{\nabla }u{|}^{2}dx)\mathrm{\Delta }u=\lambda u^3+g(u,\lambda ),\mathrm{i}\mathrm{n}\mathrm{\Omega },\\ u=0,\mathrm{o}\mathrm{n}\mathrm{\partial }\mathrm{\Omega },\end{array}$$

where $\lambda$ is a parameter. Under some natural hypotheses on $g$ and $\mathrm{\Omega }$, 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.