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Salah A. Khafagy and Z. Sadeghi, Existence of positive weak solution for a weighted system of autocatalytic reaction steady state type, Eur. J. Math. Appl. 4 (2024), Article ID 11.

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Volume 4 (2024), Article ID 11

https://doi.org/10.28919/ejma.2024.4.11

Published: 30/04/2024

Abstract:

We establish the existence results of positive weak solution for the weighted $p$-Laplacian autocatalytic reaction problem $-\Delta_{P,p}u=\lambda m(x)[\nu a(x)u^{\alpha }-\upsilon u^{\beta }]$ in $\Omega,$ $u=0$ on $\partial \Omega$, where $\Delta _{P,p}$ with $p>1$ and $P=P(x)$ is a weight function, denotes the weighted $p$-Laplacian defined by $\Delta_{P,p}u\equiv div[P(x)|\nabla u|^{p-2}\nabla u],m(x),a(x)$ are weight functions, $\lambda ,\nu ,\upsilon$ are positive parameters, $\alpha +1\leq p<\beta +1$, and $\Omega \subset R^{n}$ is a bounded domain with smooth boundary $\partial \Omega$. We establish that there exists positive constant $\lambda ^{\ast }(\Omega )$ such that the above system has a positive weak solution for $\lambda \geq \lambda ^{\ast }$. We use the method of sub-supersolutions to establish our results. How to Cite: Salah A. Khafagy and Z. Sadeghi, Existence of positive weak solution for a weighted system of autocatalytic reaction steady state type, Eur. J. Math. Appl. 4 (2024), Article ID 11. https://doi.org/10.28919/ejma.2024.4.11