Salah A. Khafagy and A. Ezzat Mohamed, Existence and stability of positive weak solutions for a class of chemically reacting systems, Eur. J. Math. Appl. 4 (2024), Article ID 2.

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Volume 4 (2024), Article ID 2

https://doi.org/10.28919/ejma.2024.4.2

Published: 09/01/2024

Abstract:

In this article, we study the existence and nonexistence results of positive weak solutions for semilinear elliptic system of the form:
\begin{equation*}
\begin{cases}
-\Delta u =\lambda a(x) [f(u,v)-\frac{1}{u^\alpha}], \quad &x\in \Omega,\\
-\Delta v =\lambda b(x) [g(u,v)-\frac{1}{v^\beta}], \quad &x\in \Omega,\\
u=0=v, \quad &x \in \partial\Omega,
\end{cases}
\end{equation*}
where $\lambda$ is a positive parameter, $\alpha$,$\beta$ $\in(0,1)$ and $\Omega\subset\mathbf{R}^n(n>1)$ is a bounded domain with smooth boundary $\partial\Omega$. Here $f, g$ are $C^1$ non-decreasing functions such that $f$, $g$: $[0,\infty)\times[0,\infty) \rightarrow [0,\infty)$; $f(u,v)>0$, $g(u,v)>0$ for $u,v>0$ and $a(x)$, $b(x)$ are $C^1$ sign-changing functions that are probably negative near the boundary. In particular, on $f(0,0)$ or $g(0,0)$ there is no any sign conditions. Our approach is based on the sub-super solutions method. Also, under some certain conditions, we study the stability and instability properties of the positive weak solution for the system under consideration.

How to Cite:

Salah A. Khafagy and A. Ezzat Mohamed, Existence and stability of positive weak solutions for a class of chemically reacting systems, Eur. J. Math. Appl. 4 (2024), Article ID 2. https://doi.org/10.28919/ejma.2024.4.2