In this work, we study the existence of one-sign solutions without signum condition for the following problem:

$$\begin{array}{r}\{\begin{array}{lll}-\mathrm{\Delta }u=\lambda a(x)f(u),x\in {\mathbb{R}}^N,& u(x)\to 0,\mathrm{a}\mathrm{s}|x|\to +\mathrm{\infty },& \end{array}\end{array}$$

where $N\ge 3$, $\lambda$ is a real parameter and $a\in C_{loc}^{\alpha }({\mathbb{R}}^N,\mathbb{R})$ for some $\alpha \in (0,1)$ is a weighted function, $f\in C^{\alpha }(\mathbb{R},\mathbb{R})$, and there exist two constants $s_2<0<s_1,$ such that $f(s_1)=f(s_2)=f(0)=0$ and $sf(s)>0$ for $s\in \mathbb{R}\mathrm{\setminus }\{s_1,0,s_2\}.$ Furthermore, we consider the exact multiplicity of one-sign solutions for above problem under more strict hypotheses. We use bifurcation techniques and the approximation of connected components to prove our main results.

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.