We prove existence and bifurcation results for a semilinear eigenvalue problem in ${ℝ}^N$ $(N\ge 2)$, where the linearization — $$ has no eigenvalues. In particular, we show that under rather weak assumptions on the coefficients $\lambda =0$ is a bifurcation point for this problem in $H^1,H^2$ and $L^p$ $(2\le p\le \infty )$.

Let ${\lambda }_1\left(Q\right)$ be the first eigenvalue of the Sturm-Liouville problem $$y^{\text{'}\text{'}}-Q\left(x\right)y+\lambda y=0,y\left(0\right)=y\left(1\right)=0,0<x<1.$$ We give some estimates for $m_{\alpha ,\beta ,\gamma }={inf}_{Q\in T_{\alpha ,\beta ,\gamma }}{\lambda }_1\left(Q\right)$ and $M_{\alpha ,\beta ,\gamma }={sup}_{Q\in T_{\alpha ,\beta ,\gamma }}{\lambda }_1\left(Q\right)$, where $T_{\alpha ,\beta ,\gamma }$ is the set of real-valued measurable on $\left[0,1\right]$ $x^{\alpha }{(1-x)}^{\beta }$-weighted $L_{\gamma }$-functions $Q$ with non-negative values such that ${\int }_0^1{x}^{\alpha }{(1-x)}^{\beta }{Q}^{\gamma }\left(x\right)\mathrm{d}x=1$ $(\alpha ,\beta ,\gamma \in ℝ,\gamma \ne 0)$.