Positive solutions for singular higher order nonlinear equations
A nonnegative real number \(\lambda\) is called an eigenvalue of the nonlinear boundary value problem \[ y^{(n)}+\lambda q(x)f(y)=0,\;y^{(i)}(0)=y^{(n-2)}(1)=0,\;0\leq i\leq n-2, \] if there is a positive solution \(y\). It is assumed that \(f:[0,\infty)\to (0,\infty)\) is continuous and strictly increasing, \(q:(0,1)\to[0,\infty)\) is continuous and not identically zero, and \((1-x)q(x)\in L^1(0,1)\). It is shown that the set of eigenvalues is an interval containing \(0\), and depending on the asymptotic behavior of \(u/f(u)\) it is shown that the interval is of the form \([0,\delta)\), \([0,\delta]\), or \([0,\infty)\) with \(\delta>0\).
eigenvalue of the nonlinear boundary value problem
scientific article; zbMATH DE number 1011025