Opuscula Math. 39, no. 5 (2019), 675-689
https://doi.org/10.7494/OpMath.2019.39.5.675

 
Opuscula Mathematica

Positive solutions for the one-dimensional p-Laplacian with nonlinear boundary conditions

D. D. Hai
X. Wang

Abstract. We prove the existence of positive solutions for the \(p\)-Laplacian problem \[\begin{cases}-(r(t)\phi (u^{\prime }))^{\prime }=\lambda g(t)f(u),& t\in (0,1),\\au(0)-H_{1}(u^{\prime }(0))=0,\\cu(1)+H_{2}(u^{\prime}(1))=0,\end{cases}\] where \(\phi (s)=|s|^{p-2}s\), \(p\gt 1\), \(H_{i}:\mathbb{R}\rightarrow\mathbb{R}\) can be nonlinear, \(i=1,2\), \(f:(0,\infty )\rightarrow \mathbb{R}\) is \(p\)-superlinear or \(p\)-sublinear at \(\infty\) and is allowed be singular \((\pm\infty)\) at \(0\), and \(\lambda\) is a positive parameter.

Keywords: \(p\)-Laplacian, semipositone, nonlinear boundary conditions, positive solutions.

Mathematics Subject Classification: 34B16, 34B18.

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  • Communicated by Jean Mawhin.
  • Received: 2019-02-18.
  • Accepted: 2019-05-26.
  • Published online: 2019-09-05.
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Cite this article as:
D. D. Hai, X. Wang, Positive solutions for the one-dimensional p-Laplacian with nonlinear boundary conditions, Opuscula Math. 39, no. 5 (2019), 675-689, https://doi.org/10.7494/OpMath.2019.39.5.675

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