Positive Solutions to Semilinear Elliptic Equations with Logistic Type Nonlinearities and Constant Yield Harvesting in ℝN

We consider a class of semilinear elliptic equations − Δ u = f(x, u) in all of ℝ N with nonlinearities of the form where λ, μ are positive parameters, a(x), h(x) are positive functions, and g(u) is a super-linearly increasing function in a more general fashion than the classical logistic term u 2. From a practical point of view, these problems can provide models for fishery or hunting management (cf. [8]) where μ h(x) denotes a harvesting term and, as such, one is interested in situations allowing the existence of positive solutions. From a mathematical point of view, these elliptic problems belong to the class of so-called semi-positone problems (cf. [2]) because the nonlinearity f(x, u) satisfies f(x, 0) < 0. Under suitable assumptions on a(x), h(x), we use variational methods to show that, for each λ > λ1 (a) (where λ1 (a) denotes the principal eigenvalue of − Δ u = λ a(x)u ∈ D 1,2(ℝ N )), there exists a positive solution decaying at infinity like O(|x|−(N−2)), provided that 0 < μ < (λ). Keywords: Elliptic equations in ℝ N HarvestingLogistic type nonlinearitiesPositive solutionsSemi-positone problemsMathematics Subject Classification: 35J2035D0535B4034B18 Acknowledgments Pavel Drábek was supported by the Research Plan of the Ministry of Education, Youth and Sports of Czech Republic, no. MSM4977751301. David Costa gratefully acknowledges the kind hospitality of the Department of Mathematics of the University of West Bohemia, Czech Republic, where the present work was initiated.