Some properties of the~spectrum of nonlinear equations of Sturm--Liouville type

A. P. Buslaev

Sbornik. Mathematics, Tome 80 (1995) no. 1, pp. 1-14

Résumé

The question is considered of the number of stationary points of the Rayleigh functional \begin{equation} R(x)=R(r,p,q,\Gamma_0,w_r,w_0,x)=\dfrac{\|x\|_{q(w_0)}}{\|x^{(r)}\|_{p(w_r^{-1})}}, \qquad x\big|_{\partial I}\in \Gamma _0, \end{equation} which make up the spectrum of the nonlinear equation of Sturm–Liouville type $(1$ \begin{equation} \begin{gathered} (-1)^{r+1}\biggl(\dfrac{(x^{(r)})_{(p)}(t)}{w_r(t)}\biggr)^{(r)}+ \lambda^q w_{0}(t)x_{(q)}(t)=0, \\ x\big|_{\partial I}\in \Gamma_0, \qquad \frac{(x^{(r)})_{(p)}}{w_r}\bigg|_{\partial I} \in \Gamma_1, \end{gathered} \end{equation} where $\bigl(h(\,\cdot\,)\bigr)_{(s)}=|h(\,\cdot\,)|^{s-1}\operatorname{sgn}(h(\,\cdot\,))$. Under various assumptions on the parameters it is proved that a solution with $n$ sign changes interior to $I=[0,1]$ is unique up to normalization.