Multiple existence of positive even solutions for a two point boundary value problem on some very narrow possible parameter set. (English) Zbl 1502.34034
The paper studies the positive even solutions of the parameter-dependent Dirichlet boundary value problem
\[
\begin{cases}
u'' + (|x|^{l} + \lambda) u^{p} = 0, \quad x\in(-1,1), \\
u(-1)=u(1)=0,
\end{cases}
\]
where \(p>1\), \(l\geq0\) and \(\lambda\geq0\) are two parameters. The authors pursue the analysis in [S. Tanaka, J. Differ. Equations 255, No. 7, 1709–1733 (2013; Zbl 1295.34040)], where the existence and the multiplicity problem is considered for \(l > 1\) and \((p-1)l \geq 4\). They first present a uniqueness result for positive even solutions for a wide range of \((l,\lambda)\) (and \(p>1\) fixed). Then, they numerically investigate the region in the plane \((l,\lambda)\) which is not covered by their previous analysis; thanks to some numerical experiments they show that the multiplicity of positive solutions occurs.
Reviewer: Guglielmo Feltrin (Udine)
MSC:
| 34B18 | Positive solutions to nonlinear boundary value problems for ordinary differential equations |
| 34B08 | Parameter dependent boundary value problems for ordinary differential equations |
| 34B15 | Nonlinear boundary value problems for ordinary differential equations |
Keywords:
uniqueness of solutions; multiple existence of solutions; Pohožaev type identity; numerical verification methodCitations:
Zbl 1295.34040Software:
kvReferences:
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