具$p$-Laplace 算子的三点边值问题正解的存在性

Existence of Positive Solutions for Three-Point Boundary Value Problem with $p$-Laplacian

中文关键词:  $p$-Laplacian  三点边值  锥  Krasnoselskii's不动点定理.

英文关键词:$p$-Laplacian  three-point boundary value problem  cone  Krasnoselskii's fixed-point theorem.

中文摘要:

      本文用锥上的Krasnoselskii's不动点定理研究了具有 $p$-Laplace算子的三点边值问题:$$\left\{\begin{array}{ll}(\phi_p(u'(t)))'+a(t)f(u(t))=0,\;\;\;t\in (0,1),\\u(0)=\alpha u(\eta),\;u(1)=\beta u(\eta),\end{array}\right.$$其中 $0<\alpha,\; \beta< 1,$ $0<\eta< 1$ 且 $\phi_p(z)=|z|^{p-2}z,\; p>1.$ 在$f$ 满足一定的增长条件下, 得到方程正解的存在性.作为应用, 给出两个例子.

英文摘要:

      By means of the Krasnoselskii's fixed-point theorem in cone, we study the existence of positive solution for the three-point boundary value problem with $p$-Laplacian operator $$\left\{\begin{array}{ll}(\phi_p(u'(t)))'+a(t)f(u(t))=0,\;\;\;t\in (0,1),\\ u(0)=\alpha u(\eta),\;u(1)=\beta u(\eta), \end{array}\right.$$ where $0<\alpha,\; \beta< 1,$ $0<\eta< 1$ and $\phi_p(z)=|z|^{p-2}z,\; p>1.$ Sufficient conditions are given which guarantee the existence of positive solutions of this problem.