Positive solutions of a fourth-order boundary value probleminvolving    derivatives of all orders

Zhilin Yang; Jingxian Sun

doi:10.3934/cpaa.2012.11.1615

Article Contents

2012, Volume 11, Issue 5: 1615-1628. Doi: 10.3934/cpaa.2012.11.1615

This issue Previous Article Lie groups related to Hörmander operators and Kolmogorov-Fokker-Planck equations Next Article Improved Caffarelli-Kohn-Nirenberg and trace inequalities for radial functions

Positive solutions of a fourth-order boundary value problem involving derivatives of all orders

Zhilin Yang^1, and
Jingxian Sun^2,

1.
Department of Mathematics, Qingdao Technological University, No 11 Fushun Road, Qingdao, Shandong Province
2.
Department of Mathematics, Xuzhou Normal University, Xuzhou 221116

Received: May 2010

Revised: December 2011

Published: September 2012

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Abstract

This paper is mainly concerned with the existence, multiplicity and uniqueness of positive solutions for the fourth-order boundary value problem \begin{eqnarray*} u^{(4)}=f(t,u,u^\prime,-u^{\prime\prime},-u^{\prime\prime\prime}),\\ u(0)=u^\prime(1)=u^{\prime\prime}(0)=u^{\prime\prime\prime}(1)=0, \end{eqnarray*} where $f\in C([0,1]\times\mathbb R_+^4,\mathbb R_+)(\mathbb R_+:=[0,\infty))$. Based on a priori estimates achieved by utilizing some integral identities and inequalities, we use fixed point index theory to prove the existence, multiplicity and uniqueness of positive solutions for the above problem. Finally, as a byproduct, our main results are applied to establish the existence, multiplicity and uniqueness of symmetric positive solutions for the fourth order Lidstone problem.

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Mathematics Subject Classification: Primary: 34B18, 47H11; Secondary: 47N20, 45J05, 45M20.

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