Existence results for a class of $p$–$q$ Laplacian semipositone boundary value problems
Let $\Omega$ be a bounded domain in $\mathbb{R}^N$; $N>1$ with a smooth boundary or $\Omega=(0,1)$. We study positive solutions to the boundary value problem of the form: \begin{equation*} \begin{aligned} -\Delta_p u - \Delta_q u&=\lambda f(u) &&\mbox{in}~\Omega,\\ u &= 0 &&...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
University of Szeged
2020-12-01
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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=8933 |
| _version_ | 1797830499602792448 |
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| author | Ujjal Das Amila Muthunayake Ratnasingham Shivaji |
| author_facet | Ujjal Das Amila Muthunayake Ratnasingham Shivaji |
| author_sort | Ujjal Das |
| collection | DOAJ |
| description | Let $\Omega$ be a bounded domain in $\mathbb{R}^N$; $N>1$ with a smooth boundary or $\Omega=(0,1)$. We study positive solutions to the boundary value problem of the form:
\begin{equation*}
\begin{aligned}
-\Delta_p u - \Delta_q u&=\lambda f(u) &&\mbox{in}~\Omega,\\
u &= 0 &&\mbox{on}~\partial\Omega,
\end{aligned}
\end{equation*}
where $q \in [2,p)$, $\lambda$ is a positive parameter, and $f:[0,\infty) \mapsto \mathbb{R}$ is a class of $C^1$, non-decreasing and $p$-sublinear functions at infinity (i.e. $\lim_{t \rightarrow \infty} \frac{f(t)}{t^{p-1}}=0$) that are negative at the origin (semipositone). We discuss the existence of positive solutions for $\lambda\gg1$. Further, when $p=4,q=2$, $\Omega=(0,1)$ and $f(s)=(s+1)^\gamma-2$; $\gamma \in (0,3)$, we provide the exact bifurcation diagram for positive solutions. In particular, we observe two positive solutions for a finite range of $\lambda$ and a unique positive solution for $\lambda\gg1.$ |
| first_indexed | 2024-04-09T13:37:06Z |
| format | Article |
| id | doaj.art-400a1f31f1254912aefab4f94eb7bfcf |
| institution | Directory Open Access Journal |
| issn | 1417-3875 |
| language | English |
| last_indexed | 2024-04-09T13:37:06Z |
| publishDate | 2020-12-01 |
| publisher | University of Szeged |
| record_format | Article |
| series | Electronic Journal of Qualitative Theory of Differential Equations |
| spelling | doaj.art-400a1f31f1254912aefab4f94eb7bfcf2023-05-09T07:53:10ZengUniversity of SzegedElectronic Journal of Qualitative Theory of Differential Equations1417-38752020-12-012020881710.14232/ejqtde.2020.1.888933Existence results for a class of $p$–$q$ Laplacian semipositone boundary value problemsUjjal Das0Amila Muthunayake1Ratnasingham Shivaji2The Institute of Mathematical Sciences, HBNI, Chennai, IndiaDepartment of Mathematics and Statistics, University of North Carolina at Greensboro, Greensboro, NC, U.S.A.Department of Mathematics and Statistics, The University of North Carolina at Greensboro, Greensboro, NC, U.S.A.Let $\Omega$ be a bounded domain in $\mathbb{R}^N$; $N>1$ with a smooth boundary or $\Omega=(0,1)$. We study positive solutions to the boundary value problem of the form: \begin{equation*} \begin{aligned} -\Delta_p u - \Delta_q u&=\lambda f(u) &&\mbox{in}~\Omega,\\ u &= 0 &&\mbox{on}~\partial\Omega, \end{aligned} \end{equation*} where $q \in [2,p)$, $\lambda$ is a positive parameter, and $f:[0,\infty) \mapsto \mathbb{R}$ is a class of $C^1$, non-decreasing and $p$-sublinear functions at infinity (i.e. $\lim_{t \rightarrow \infty} \frac{f(t)}{t^{p-1}}=0$) that are negative at the origin (semipositone). We discuss the existence of positive solutions for $\lambda\gg1$. Further, when $p=4,q=2$, $\Omega=(0,1)$ and $f(s)=(s+1)^\gamma-2$; $\gamma \in (0,3)$, we provide the exact bifurcation diagram for positive solutions. In particular, we observe two positive solutions for a finite range of $\lambda$ and a unique positive solution for $\lambda\gg1.$http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=8933$p$–$q$ laplaciansemipositone problemspositive solutions |
| spellingShingle | Ujjal Das Amila Muthunayake Ratnasingham Shivaji Existence results for a class of $p$–$q$ Laplacian semipositone boundary value problems Electronic Journal of Qualitative Theory of Differential Equations $p$–$q$ laplacian semipositone problems positive solutions |
| title | Existence results for a class of $p$–$q$ Laplacian semipositone boundary value problems |
| title_full | Existence results for a class of $p$–$q$ Laplacian semipositone boundary value problems |
| title_fullStr | Existence results for a class of $p$–$q$ Laplacian semipositone boundary value problems |
| title_full_unstemmed | Existence results for a class of $p$–$q$ Laplacian semipositone boundary value problems |
| title_short | Existence results for a class of $p$–$q$ Laplacian semipositone boundary value problems |
| title_sort | existence results for a class of p q laplacian semipositone boundary value problems |
| topic | $p$–$q$ laplacian semipositone problems positive solutions |
| url | http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=8933 |
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