Entire solutions of the nonlinear eigenvalue logistic problem with sign-changing potential and absorbtion

We are concerned with positive solutions decaying to zero at infinity for the logistic equation $-\mathrm{\Delta }u=\lambda (V(x)u-f(u))$ in ${\text{RR}}^N$, where $V(x)$ is a variable potential that may change sign, $\lambda$ is a real parameter, and $f$ is an absorbtion term such that the mapping $f(t)\mathrm{/}t$ is increasing in $(0,\mathrm{\infty })$. We prove that there exists a bifurcation non-negative number $\mathrm{\Lambda }$ such that the above problem has exactly one solution if $\lambda

\Lambda$,butnosuchasolutionexistsprovided$\lambda\leq\Lambda$.