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

Abstract

We are concerned with positive solutions decaying to zero at infinity for the logistic equation $-Δu=λ(V(x)u-f(u))$ in $\RR^N$, where $V(x)$ is a variable potential that may change sign, $λ$ is a real parameter, and $f$ is an absorbtion term such that the mapping $f(t)/t$ is increasing in $(0,\infty)$. We prove that there exists a bifurcation non-negative number $Λ$ such that the above problem has exactly one solution if $λ>Λ$, but no such a solution exists provided $λ\leqΛ$.