Exact Multiplicity of Positive Solutions for a Class of Second-Order Two-Point Boundary Problems with Weight Function

• Main Authors: Luo Hua, An Yulian
• Format: Article
• Language: English
• Published: SpringerOpen 2010-01-01
• Series: Boundary Value Problems
• Online Access: http://www.boundaryvalueproblems.com/content/2010/207649

<p/> <p>An exact multiplicity result of positive solutions for the boundary value problems <inline-formula> <graphic file="1687-2770-2010-207649-i1.gif"/></inline-formula>, <inline-formula> <graphic file="1687-2770-2010-207649-i2.gif"/></inline-formula>, <inline-formula> <graphic file="1687-2770-2010-207649-i3.gif"/></inline-formula>, <inline-formula> <graphic file="1687-2770-2010-207649-i4.gif"/></inline-formula> is achieved, where <inline-formula> <graphic file="1687-2770-2010-207649-i5.gif"/></inline-formula> is a positive parameter. Here the function <inline-formula> <graphic file="1687-2770-2010-207649-i6.gif"/></inline-formula> is <inline-formula> <graphic file="1687-2770-2010-207649-i7.gif"/></inline-formula> and satisfies <inline-formula> <graphic file="1687-2770-2010-207649-i8.gif"/></inline-formula>, <inline-formula> <graphic file="1687-2770-2010-207649-i9.gif"/></inline-formula> for <inline-formula> <graphic file="1687-2770-2010-207649-i10.gif"/></inline-formula> for some <inline-formula> <graphic file="1687-2770-2010-207649-i11.gif"/></inline-formula>. Moreover, <inline-formula> <graphic file="1687-2770-2010-207649-i12.gif"/></inline-formula> is asymptotically linear and <inline-formula> <graphic file="1687-2770-2010-207649-i13.gif"/></inline-formula> can change sign only once. The weight function <inline-formula> <graphic file="1687-2770-2010-207649-i14.gif"/></inline-formula> is <inline-formula> <graphic file="1687-2770-2010-207649-i15.gif"/></inline-formula> and satisfies <inline-formula> <graphic file="1687-2770-2010-207649-i16.gif"/></inline-formula>, <inline-formula> <graphic file="1687-2770-2010-207649-i17.gif"/></inline-formula> for <inline-formula> <graphic file="1687-2770-2010-207649-i18.gif"/></inline-formula>. Using bifurcation techniques, we obtain the exact number of positive solutions of the problem under consideration for <inline-formula> <graphic file="1687-2770-2010-207649-i19.gif"/></inline-formula> lying in various intervals in <inline-formula> <graphic file="1687-2770-2010-207649-i20.gif"/></inline-formula>. Moreover, we indicate how to extend the result to the general case.</p>