一类一阶周期边值问题多个正解的存在性

本文研究了一阶周期边值问题■多个正解的存在性,其中λ>0是一个参数,a∈C（R,0,∞）)是一个T-周期函数且∫^T₀a（t）dt>0,f∈C（0,∞）,（0,∞）)且单调递增.在■的条件下,本文证明存在一个λ^*>0,使当0<λ<λ^*时问题不存在正解;当λ=λ^*时问题至少存在一个正解;当λ>λ^*时问题至少存在两个正解.主要结果的证明基于上下解方法和Leray-Schauder度.

Existence of multiple positive solutions for a class of first-order periodic boundary value problems

We investigate the existence and the multiplicity of positive solutions for first-order periodic boundary value problems $$ \left\{\begin{array}{ll} -u''(t)+a(t)u(t)=\lambda f(u(t)), \ \ \ 00$ is a parameter,~$a\in C(\mathbb{R},0,\infty))$ is a~$T$-periodic function with $\int^T_0a(t)dt>0$,~$f\in C(0,\infty),(0,\infty))$. With $f_{0}=\lim\limits_{u\rightarrow0^{+}}\frac{f(u)}{u}=0$,~$f_{\infty}=\lim\limits_{u\rightarrow\infty}\frac{f(u)}{u}=0$ and $f$ is a monotonically increasing function,~we show that there exists a $\lambda^*>0$ such that the problem~(P)~has no positive solution,~at least one positive solution and at least two positive solutions,~for $0<\lambda<\lambda^*$,~$\lambda=\lambda^*$, $\lambda>\lambda^*$, respectively. The proof of the main results are based on upper and lower solutions and Leray-Schauder degree.