Study on nonlinear elliptic boundary value problems with Allee effects, arising in population dynamics

利用变分技术和分岔理论，证明了种群动力学中非线性椭圆型边值问题正解的存在性和多重性。特别地，我们得到了一类正解的分岔，这表明分岔分量来源于种群动态的Allee效应，暗示了物种的条件持续性。通过考虑相关的线性化特征值问题的正主特征值，讨论了分岔点与所含系数的关系，并给出了分岔点爆破的充分必要条件。

We prove the existence and multiplicity of positive solutions of nonlinear elliptic boundary value problems arising in population dynamics, by using a variational technique and the bifurcation theory. Especially, we obtain some type of bifurcation of positive solutions, which suggests that the bifurcation component is derived from the Allee effect from population dynamics, implying a conditional persistence of species. We also discuss the dependence of the bifurcation point on coefficients included in the problem and give necessary and sufficient conditions for the blowing-up of the bifurcation point, by considering the positive principal eigenvalue of the associated, linearized eigenvalue problem.