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Some properties of the spectrum of nonlinear equations of Sturm–Liouville type
A. P. Buslaev
Abstract:
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
$(1<p,q<\infty)$
\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.
UDC:
517.5
MSC: Primary
34B24,
34B15,
34L05; Secondary
41A55,
46E35 Received: 25.05.1992
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