This paper investigates the existence and uniqueness of one positive solution to a fractional p-Laplacian equation with logistic-type nonlinearity, given by \(\begin{aligned} \left( -\Delta \right) ^s_p u - \alpha (x) u^{p-1} + \beta (x) u^p = \lambda m(x) u^{p-1} \end{aligned}\) in a smooth bounded domain \(\Omega \subset \mathbb {R}^N\) , with \(u = 0\) in \(\mathbb {R}^N {\setminus } \Omega \) . Arising in population dynamics models with harvesting and variable reproduction rates, the problem is analyzed using the anti-maximum principle and the generalized principal eigenvalue of the operator \(\mathcal {L}_\Phi [\psi ] = \left( -\Delta \right) _p^s \psi + \Phi \psi ^{p-1}\) , where \(\Phi \in L^\infty (\Omega )\) . Existence is established via subsolution and supersolution techniques, while uniqueness is proven through a comparison principle, for \(N \ge 2\) . The method of proof relies on Picone-type inequalities in combination with classical minimization procedures to show the equivalence between the two types of eigenvalues and the existence of the associated first eigenfunction. Moreover, the anti-maximum principle is proved and then applied to obtain a solution to the problem.
