Asymptotic formula of eigenvalues of sublinear Sturm-Liouville problems (Q1343061)

The sublinear Sturm-Liouville problem with power nonlinearity \(-u''(x) + | u(x) |^{p-1} u(x) = \lambda u(x)\), \(x \in I = (0,1)\), \(u(0) = u(1) = 0\) is considered. Asymptotic formulas of eigenvalues \(\lambda\) are established as \(\alpha_{n, \lambda} \to 0\) and \(\sigma_{n, \lambda} \to 0\) with the remainder estimates, \(\alpha_{n, \lambda} : = \| u_{n, \lambda} \|_{L^ 2}\), \(\sigma_{n, \lambda} : = \| u_{n, \lambda} \|_{L^ \infty}\) being the \(L^ 2\)-norm and \(L^ \infty\)-norm, respectively. At first the ground state \(w\) of an equation corresponding to the above problem is studied, then formulas are proved for \(n=1\) and finally for \(n \geq 2\).