一类半正二阶Dirichlet系统正解的存在性

Existence of Positive Solutions fora Class of Semipositone Second-Order Dirichlet Systems

摘要: 本文研究一类半正二阶 Dirichlet 系统$$\{\begin{array}{ll}-u^{″}(t)=\lambda a(t)f(u(t),v(t)),& t\in (0,1),\\ -v^{″}(t)=\lambda b(t)g(u(t),v(t)),& t\in (0,1),\\ u(0)=u(1)=0,\\ v(0)=v(1)=0\end{array}$$正解的存在性，其中λ > 0是参数,a, b∈ L¹(0, 1), f, g∈C([0,+oo) × [O,+oo), ℝ)且存在q∈ L¹(0, 1),使得 f(u(t), v(t) ≥一q(t), g(u(t), v(t) ≥ 一q(t). 证得存在λ₀ > 0,使得当0 < λ < λ₀ 时, 该问题至少存在一个正解。主要结果的证明基于Krasnosel'skii型不动点定理。

Abstract: We are concerned with the existence of positive solutions for a class of semipositone second-order Dirichlet systems: $$\{\begin{array}{ll}-u^{″}(t)=\lambda a(t)f(u(t),v(t)),& t\in (0,1),\\ -v^{″}(t)=\lambda b(t)g(u(t),v(t)),& t\in (0,1),\\ u(0)=u(1)=0,\\ v(0)=v(1)=0,\end{array}$$ where λ > 0 is a parameter, a, b∈ L¹(0, 1), f; g∈C([0,+oo) × [O,+oo), ℝ) such that there exists q∈ L¹(0, 1) satisfying f(u(t), v(t) ≥一q(t) and g(u(t), v(t) ≥ 一q(t). We prove that there exists λ₀ > 0 such that the problem has at least one positive solution for 0 < λ < λ₀. The proof of the main result is based on the Krasnosel'skii type fixed pointtheorem.