一类 Kirchhoff 方程半正问题正解的存在性

The Existence of Positive Solutionsfor a Class of Kirchhoff EquationSemipositone Problem

摘要: 考察了下列边值问题$$\{\begin{array}{l}-(\alpha +\beta \Vert u{\Vert }^2)u^{″}=\lambda f(t,u),t\in (r,R),\\ u(r)=u(R)=0\end{array}$$正解的存在性，其中 λ 为正参数，α > 0, β ≥ 0, f : [r, R] × [0, ∞) → ℝ为连续函数，且存在 M > 0 使得当 t ∈ [r, R] 时，f (t, u) ≥ −M。 运用 Krasnoselskii’s 不动点定理证明存在常数 λ₀ > 0，使得当 0 < λ < λ₀时， 该问题存在一个正解。

Abstract: We are concerned with the existence of positive solutions for boundary value problem $$\{\begin{array}{l}-(\alpha +\beta \Vert u{\Vert }^2)u^{″}=\lambda f(t,u),t\in (r,R),\\ u(r)=u(R)=0\end{array}$$where λ > 0 is a parameter, α > 0, β ≥ 0, f : [r;R] × [0, ∞)→ ℝ is a continuous function and exists an M > 0 such that f(t, u) ≥ −M for t ∈ [r, R]. By applying Krasnoselskii's fixed point theorem, we proved that there exists λ₀ > 0 such that the problem has a positive solution for 0 < λ < λ₀.