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

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  • We are concerned with positive solutions decaying to zero at infinity for the logistic equation $-\Delta u=\lambda ( V(x)u-f(u))$ in $\mathbb R^N$, 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 $f(t)/t$ is increasing in $(0,\infty)$. We prove that there exists a bifurcation non-negative number $\Lambda$ such that the above problem has exactly one solution if $\lambda >\Lambda$, but no such a solution exists provided $\lambda\leq\Lambda$.
    Mathematics Subject Classification: 35A05, 35B40, 35J60, 37K50, 92D25.


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