Bifurcation and one-sign solutions for semilinear elliptic problems in ${\mathbb{R}}^N$

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.