Exact Multiplicity of Positive Solutions for a Class of Semilinear Problem, II

Combined Effects of Singular and Superlinear Nonlinearities in Some Singular Boundary Value Problems

Multiplicity and asymptotic behavior of positive solutions for a singular semilinear elliptic problem

We study the positive steady state distributions and dynamical behavior of reaction-diffusion equation with weak Allee effect type growth, in which the growth rate per capita is not monotonic as in logistic type, and the habitat is assumed to be a heterogeneous bounded region. The existence of multiple steady states is shown, and the global bifurcation diagrams are obtained. Results are applied to a reaction-diffusion model with type II functional response, and also a model with density-dependent diffusion of animal aggregation.

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Persistence in reaction diffusion models with weak allee effect

We consider a reaction-diffusion equation which models the constant yield harvesting to a spatially heterogeneous population which satisfies a logistic growth. We prove the existence, uniqueness and stability of the maximal steady state solutions under certain conditions, and we also classify all steady state solutions under more restricted conditions. Exact global bifurcation diagrams are obtained in the latter case. Our method is a combination of comparison arguments and bifurcation theory.

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Diffusive logistic equation with constant yield harvesting, I: Steady states

We consider the singular boundary value problemWe study the existence, uniqueness, regularity and the dependency on parameters of the positive solutions under various assumptions.

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On a singular nonlinear semilinear elliptic problem

We study an elliptic boundary-value problem with singular nonlinearity via the method of monotone iteration scheme: −∆u(x) = f(x, u(x)), x ∈ Ω, u(x) = φ(x), x ∈ ∂Ω, where ∆ is the Laplacian operator, Ω is a bounded domain in RN , N ≥ 2, φ ≥ 0 may take the value 0 on ∂Ω, and f(x, s) is possibly singular near s = 0. We prove the existence and the uniqueness of positive solutions under a set of hypotheses that do not make neither monotonicity nor strict positivity assumption on f(x, s), which improvements of some previous results.

Miaoxin Yao

Papers

On a singular nonlinear semilinear elliptic problem

Positive solutions for elliptic equations with singular nonlinearity.