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This article is cited in 1 scientific paper (total in 1 paper)
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.
Received: 25.05.1992
Citation:
A. P. Buslaev, “Some properties of the spectrum of nonlinear equations of Sturm–Liouville type”, Mat. Sb., 184:9 (1993), 3–20; Russian Acad. Sci. Sb. Math., 80:1 (1995), 1–14
Linking options:
https://www.mathnet.ru/eng/sm1010https://doi.org/10.1070/SM1995v080n01ABEH003511 https://www.mathnet.ru/eng/sm/v184/i9/p3
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Abstract page: | 342 | Russian version PDF: | 96 | English version PDF: | 9 | References: | 61 | First page: | 1 |
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