Positive solutions of fourth order singular boundary value problems (Q2763910)

Sufficient conditions for the existence of \(C^{2}[0,1]\) positive solutions to singular boundary value problems for fourth-order differential equations are obtained. Exactly, the authors consider the following problem NEWLINE\[NEWLINEx^{(4)}(t)=\lambda a(t)f(t,x(t)),\;0<t<1,\quad x(0)=x(1)=0,\quad x^{\prime \prime }(0)=x^{\prime \prime }(1)=0,\tag{1}NEWLINE\]NEWLINE where \(\lambda >0\) is a real parameter, \(a(t)\) and \(f(t,x)\) satisfy the following conditions: NEWLINENEWLINENEWLINE\((H_{1})\) \(a(t)\in C((0,1),[0,\infty))\), \(a(t)\neq 0\) in \((0,1)\) and the following inequalities hold NEWLINE\[NEWLINE\begin{matrix} \int_{0}^{1}t(1-t)a(t)dt &\in &(0,\infty) \\ \lim_{t\rightarrow 0^{+}}t\int_{t}^{1}(1-s)a(s)ds &=&0 \\ \lim_{t\rightarrow 1^{-}}(1-t)\int_{0}^{t}sa(s)ds &=&0 \end{matrix}\tag{*}NEWLINE\]NEWLINE NEWLINENEWLINENEWLINE\((H_{2})\) \(f(t,x)\in C([0,1]\times [0,\infty),(0,\infty))\) is nondecreasing in \(x\).NEWLINENEWLINENEWLINEThe main results are:NEWLINENEWLINENEWLINETheorem 1: Assume that \((H_{1})-(H_{2})\) hold. Then there exists \( \lambda _{1}>0\) such that problem \((1)\) has at least one \(C^{2}[0,1]\) positive solution for \(0<\lambda \leq \lambda_{1}\).NEWLINENEWLINENEWLINETheorem 2: Assume that \((H_{1})-(H_{2})\) hold. Furthermore, \(f\) satisfies the following condition: NEWLINENEWLINENEWLINE\((H_{3})\) there exists \(d>0\) such that \(f(t,x)\geq dx\) for \(0\leq t\leq 1,\) \(x>0.\) Then there exists \(\lambda _{2}>0\) such that problem \((1)\) has at least one \(C^{2}[0,1]\) positive solution for \(0<\lambda < \lambda_{2}\) while there is no such solution for \(\lambda >\lambda _{2}\).NEWLINENEWLINENEWLINEThe proofs of the results are obtained by means of the method of lower and upper solutions with the maximum principle.