Precise spectral asymptotics for the Dirichlet problem \(-u''(t)+g(u(t))=\lambda\sin u(t)\) (Q1599115)

Here, Dirichlet problems for nonlinear differential equations of the form \[ -u''(t)+g(u(t))= \lambda \sin u(t), \quad u(t) > 0, \] on an interval \((-T,T)\) are considered with \(\lambda, T > 0\). The main result concerns the relation between the parameter \(\lambda\) and the boundary layer \(u(\lambda)\) when \(\lambda \gg 1\). To this end, a solution pair \((\lambda(\varepsilon),u_\varepsilon)\) is parametrized by \(0 < \varepsilon < T\). Under the assumptions that \(g\in C^1({\mathbb{R}})\), \(g\) is odd in \(u\), \(g(u) > 0\) for \(u>0\), \(g(0)=g'(0)=0\) and \(g(u)/u\) is strictly increasing for \(u>0\), the author establishes precise asymptotic formulas \( \lambda(\varepsilon)=1/\varepsilon^2 + C_1 + o(1)\) with exact second term as \(\varepsilon \to 0\), explicitly influenced by the nonlinearity \(g\).