On spectral asymptotics for nonlinear Sturm-Liouville problems
The author considers the following nonlinear Sturm-Liouville problem \[ -u''(x)= f(u(x))-\mu u(x),\quad x\in I=(0,1), \qquad u(0)=u(1)=0, \] where \(f(u)\) is odd and \(C^1\). He defines \(\alpha_n(\mu)\) by \[ \alpha_n(\mu)= \Biggl(\int_I|u_n(\mu,x) |^2 dx\Biggr)^{1/2}, \] where \((u_n(\mu,x),\mu)\in C^2(I)\times (-(n\pi)^2,\infty)\) is a solution of the problem, and \(n-1\) is the number of the interior zeros of \(u_n(\mu,x)\). Under certain conditions on \(f\) the following asymptotic formula for \(\alpha_n(\mu)\) is established \((\mu\to\infty)\) \[ \alpha_n(\mu)^2= B_n(g^{-1}(\mu))^2 \mu^{-1/2}+ o((g^{- 1}(\mu))^2 \mu^{-1/2}), \] where \(g(t)= f(t)/t\), \(g^{-1}\) is the inverse function of \(g\) and \[ B_n= n\sqrt{\pi} \frac{2}{p-1} \biggl(\frac{p+1}{2} \biggr)^{2/(p-1)} \Gamma \biggl(\frac{2}{p- 1}\biggr) \Biggl(\Gamma\biggl( \frac{p+3}{2(p-1)} \biggr)\Biggr)^{-1}. \] An advantage of the paper is that the conditions imposed on \(f\) are rather general. The author has given some typical examples of functions satisfying these conditions: \[ f(u)=|u|^{p-1} u; \quad f(u)=|u|^{p-1} u\log (|u|+1); \quad f(u)=|u|^{p-1} ue^{-|u|/(|u|+1)}. \]
scientific article; zbMATH DE number 954720