On estimates for the minimum eigenvalue for the column stability problem (Q1358171)

The so-called Lagrange problem of column stability is considered, i.e. \[ (p(x)y'')''+\lambda y''=0,\quad y(0)=y(1)=y'(0)=y'(1)=1\tag{1} \] where \(p(x)\) is a bounded measurable function such that \(\inf p(x)>0\) and \[ \int_0^1p(x)^\alpha dx=1,\quad\alpha\neq 0. \] If \(\lambda_1>0\) denotes the first eigenvalue of \((1)\) then the following assertions hold: (i) for \(\alpha >-1/2\), \(\lambda_1\) has an upper bound \(C(\alpha)\); (ii) for \(\alpha\leq-1/2\), \(\lambda_1\) may assume arbitrarily large values; (iii) for \(\alpha>-1\), \(\lambda_1\) may assume arbitrarily small positive values; (iv) for \(\alpha\leq-1\), \(\lambda_1\) has a lower bound \(C(\alpha)\). For \(\alpha>1\) an exact formula for \(C(\alpha)\) is obtained.