Global structure of positive solutions for \(p\)-Laplacian Neumann problem with indefinite weight. (English) Zbl 07846908
Summary: We are concerned with the global structure of positive solutions for \(p\)-Laplacian Neumann problem:
\[
\begin{cases}
-(\varphi_p(u^\prime))^\prime = \lambda h(x)g(u), x\in(0, 1)\\
u^\prime(0) = u^\prime(1) = 0,
\end{cases}\tag{\(P\)}
\]
where \(\varphi_p(s) = |s|^{p-2}s\), \(p > 1\), \(\lambda > 0\) is a parameter, \(h: [0, 1]\rightarrow\mathbb{R}\) is a continuous function with \(\int_0^1h(x)\mathrm{d}x < 0\), \(g: [0, \infty)\rightarrow[0, \infty)\) is a continuous function satisfying \(\lim_{s\rightarrow 0}g(s)/\varphi_p(s) = 0\) and \(\lim_{s\rightarrow\infty}g(s)/\varphi_p(s) = 0\). We obtain a \(\subset\)-shaped component of positive solutions of problem (\(P\)) provided suitable conditions. That is, there exist \(\lambda^\ast > \lambda_\ast > 0\), such that the problem (\(P\)) has two positive solutions for \(\lambda > \lambda^\ast\) and no positive solution for \(\lambda < \lambda_\ast\). The proof of main result is based upon bifurcation technology. In addition, to prove the main result, we investigate the principal eigenvalue of auxiliary problem:
\[
\begin{cases}
-(\varphi_p(u^\prime))^\prime + \frac{1}{m}\varphi_p(u) = \lambda h(x)\varphi_p(u), x\in (0,1),\\
u^\prime(0) = u^\prime(1) = 0,
\end{cases}
\]
where \(m\in\mathbb{N}^+\).
MSC:
34B18 | Positive solutions to nonlinear boundary value problems for ordinary differential equations |
34B09 | Boundary eigenvalue problems for ordinary differential equations |
34C23 | Bifurcation theory for ordinary differential equations |
34L15 | Eigenvalues, estimation of eigenvalues, upper and lower bounds of ordinary differential operators |
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