Positive solutions for concave-convex type problems for the one-dimensional $\phi$-Laplacian
Let $\Omega=(a,b)\subset\mathbb{R}$, $0\leq m,n\in L^{1}(\Omega)$, $\lambda,\mu>0$ be real parameters, and $\phi: \mathbb{R}\rightarrow\mathbb{R}$ be an odd increasing homeomorphism. In this paper we consider the existence of positive solutions for problems of the form \[% \begin{cases}...
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| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
University of Szeged
2024-09-01
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| Series: | Electronic Journal of Qualitative Theory of Differential Equations |
| Subjects: | |
| Online Access: | http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=11175 |
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| Summary: | Let $\Omega=(a,b)\subset\mathbb{R}$, $0\leq m,n\in L^{1}(\Omega)$, $\lambda,\mu>0$ be real parameters, and $\phi: \mathbb{R}\rightarrow\mathbb{R}$ be an odd increasing homeomorphism. In this paper we consider the existence of positive solutions for problems of the form
\[%
\begin{cases}
-\phi\left( u^{\prime}\right) ^{\prime}=\lambda m(x)f(u)+\mu n(x)g(u) &
\text{ in }\Omega,\\
u=0 & \text{ on }\partial\Omega,
\end{cases} \] where $f,g:[0,\infty)\rightarrow\lbrack0,\infty)$ are continuous functions which are, roughly speaking, sublinear and superlinear with respect to $\phi$, respectively. Our assumptions on $\phi$, $m$ and $n$ are substantially weaker than the ones imposed in previous works. The approach used here combines the Guo–Krasnoselskiĭ fixed-point theorem and the sub-supersolutions method with some estimates on related nonlinear problems. |
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| ISSN: | 1417-3875 |