We determine precise existence and multiplicity results for radial solutions of the Liouville-Bratu-Gelfand problem associated with a class of quasilinear radial operators, which includes perturbations of k-Hessian and p-Laplace operators. # 2002 Elsevier Science (USA)
Water resources worldwide require management to meet industrial, agricultural, and urban consumption needs. Management actions change the natural flow regime, which impacts the river ecosystem. Water managers are tasked with meeting water needs while mitigating ecosystem impacts. We develop process-oriented advection-diffusion-reaction equations that couple hydraulic flow to population growth, and we analyze them to assess the effect of water flow on population persistence. We present a new mathematical framework, based on the net reproductive rate R0 for advection-diffusion-reaction equations and on related measures. We apply the measures to population persistence in rivers under various flow regimes. This work lays the groundwork for connecting R0 to more complex models of spatially structured and interacting populations, as well as more detailed habitat and hydrological data.
To fully understand population persistence in river ecosystems, it is necessary to consider the effect of the water flow, which varies tremendously with seasonal fluctuations of water runoff and snow melt. In this paper, we study integrodifference models for growth and dispersal in the presence of advective flow with both periodic (alternating) and random kernel parameters. For the alternating kernel model, we obtain the principal eigenvalue of the linearization operator to determine population persistence and derive a boundary value problem to calculate it. For the random model, we establish two persistence metrics: a generalized spectral radius and the asymptotic growth rate, which are mathematically equivalent but can be understood differently, to determine population persistence or extinction. The theoretical framework and methods for calculations are provided, and the framework is applied to calculating persistence in highly variable river environments.Y. J. and M. A. L.
This problem was studied by Liouville 30] in the case n = 1, Bratu 4] in the case n = 2, and later, Gelfand 13] for higher dimensions. Of particular interest is the relationship between the space dimension and multiplicity results for (0.6) rst observed by Joseph and Lundgren 20]. The results may be divided up into three cases, which we now brie y recall: Case I: n = 1; 2. There exists a > 0 such that (0.6) has exactly one solution for = and exactly two solutions for 0 < <. Case II: 3 n 9. The continuum of solutions to (0.6) oscillates around = 2(n ? 2), with the amplitude of oscillations tending to zero, as kuk ! 1. Case III: n 10. Equation (0.6) has a unique solution for each 2 (0; 2(n ? 2)) and no solutions for 2(n ? 2). In 9], the authors consider the Liouville-Gelfand problem associated with (0.5) for the k-Hessian operator when k = n=2. The purpose of this paper is to demonstrate how results concerning equations of the form (0.2) may be established using topological methods. In particular, using recent results due to Trudinger and Wang 43, 44, 45] for k-Hessian operators, we shall study (0.2) from the perspective of global bifurcation.
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