Electron. J. Differential Equations, Vol. 2018 (2018), No. 92, pp. 1-14.

Positive solutions for the one-dimensional Sturm-Liouville superlinear p-Laplacian problem

Khanh Duc Chu, Dang Dinh Hai

Abstract:
We prove the existence of positive classical solutions for the p-Laplacian problem
$$\displaylines{ -(r(t)\phi (u'))'=f(t,u),\quad t\in (0,1), \cr au(0)-b\phi ^{-1}(r(0))u'(0)=0,\ cu(1)+d\phi ^{-1}(r(1))u'(1)=0, }$$
where $\phi (s)=|s|^{p-2}s$, $p>1$, $f:(0,1)\times [ 0,\infty )\to\mathbb{R}$ is a Caratheodory function satisfying
$$ \limsup_{z\to 0^{+}} \frac{f(t,z)}{z^{p-1}}<\lambda_1 <\liminf_{z\to \infty }\frac{f(t,z)}{z^{p-1}} $$
uniformly for a.e. $t \in (0,1)$, where $\lambda _1$ denotes the principal eigenvalue of $-(r(t)\phi (u'))'$ with Sturm-Liouville boundary conditions. Our result extends a previous work by Manasevich, Njoku, and Zanolin to the Sturm-Liouville boundary conditions with more general operator.

Submitted February 12, 2018. Published April 17, 2018.
Math Subject Classifications: 34B15, 34B18.
Key Words: p-Laplacian; superlinear; positive solutions.

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Khanh Duc Chu
Faculty of Mathematics and Statistics
Ton Duc Thang University
Ho chi Minh City, Vietnam
email: chuduckhanh@tdt.edu.vn
Dang Dinh Hai
Department of Mathematics and Statistics
Mississippi state University
Mississippi State, MS 39762, USA
email: dang@math.msstate.edu

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