Uniqueness for a class of \(p\)-Laplacian problems when the reaction term tends to zero at infinity. (English) Zbl 1464.34048
The authors consider a boundary value problem with \(p\)-Laplacian:
\[
\begin{cases}
-(\phi(u'))'=\lambda h(t) f(u),~~ t\in (0,1),\\
u(0)=u(1)=0,
\end{cases}\tag{1}
\]
where \(\phi(s)=|s|^{p-2}s\), \(p>1\), \(h: (0,1)\to (0,\infty)\), \(f: (0,\infty)\to \mathbb{R}\) is a differentiable function and \(\lambda\) is a positive parameter.
Sufficient conditions are given for the uniqueness and asymptotic behavior of positive solutions to (1) for \(\lambda\) large when \(f(u) \sim u^{-\gamma}\) at \(\infty\) for some \(\gamma\in (0,1)\) with \(\lim_{u\to 0^{+}}f(u) = -\infty\), and \(h\) is a decreasing function with possible singularity at \(t = 0\). The case when \(h\) is increasing with possible singularity at \(t = 1\) is also discussed.
These results are complement the results in [K. D. Chu et al., J. Math. Anal. Appl. 472, No. 1, 510–525 (2019; Zbl 1418.35196)].
Sufficient conditions are given for the uniqueness and asymptotic behavior of positive solutions to (1) for \(\lambda\) large when \(f(u) \sim u^{-\gamma}\) at \(\infty\) for some \(\gamma\in (0,1)\) with \(\lim_{u\to 0^{+}}f(u) = -\infty\), and \(h\) is a decreasing function with possible singularity at \(t = 0\). The case when \(h\) is increasing with possible singularity at \(t = 1\) is also discussed.
These results are complement the results in [K. D. Chu et al., J. Math. Anal. Appl. 472, No. 1, 510–525 (2019; Zbl 1418.35196)].
Reviewer: Sotiris K. Ntouyas (Ioannina)
MSC:
| 34B18 | Positive solutions to nonlinear boundary value problems for ordinary differential equations |
| 34B40 | Boundary value problems on infinite intervals for ordinary differential equations |
| 34B16 | Singular nonlinear boundary value problems for ordinary differential equations |
| 34B08 | Parameter dependent boundary value problems for ordinary differential equations |
Citations:
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