European Journal of Mathematics and Applications

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Volume 6 (2026), Article ID 3

https://doi.org/10.28919/ejma.2026.6.3

Published: 05/02/2026

Abstract:

In the present article, we investigate the stability results concerning the weak solutions of the following $(p,q)$-Laplacian system
\begin{equation*}
\left.
\begin{array}{cc}
-\Delta _{p}u+\lambda _{p}|u|^{p-2}u=a(x)f(u,v) & \text{in}\,\ \Omega , \\
-\Delta _{q}v+\lambda _{q}|v|^{q-2}v=b(x)g(u,v) & \text{in}\,\ \Omega , \\
\Sigma u=0=\Sigma v & \text{on}\,\ \ \partial \Omega .
\end{array}
\right\}
\end{equation*}
where $\Delta _{p}u\equiv div[|\nabla u|^{p-2}\nabla u]$, with $p>1$, denotes the $p$-Laplacian operator. Here $\lambda _{p},\lambda _{q}$ are positive parameters, $a(x),b(x)$ are continuous functions from $\Omega$ to $\mathbf{R}$ and $f,g:[0,\infty )\times \lbrack 0,\infty )\rightarrow \mathbf{R}$ are $ C^{1}$ functions. $\Omega \subset \mathbf{R}^{n}$ is a bounded domain with smooth boundary $\Sigma u=\rho l(x)u+(1-\rho)\frac{\partial u}{\partial n}$ where $\rho \in \lbrack 0,1]$, $l:\partial \Omega \rightarrow \mathbf{R}^{+}$ where $l=1$ when $\rho =1$. Under certain conditions, we establish that every positive weak solution is either stable or unstable.

How to Cite:

Salah A. Khafagy, Z. Sadeghi, Stability results of positive solutions for (p, q)-Laplacian system with applications, Eur. J. Math. Appl. 6 (2026), Article ID 3. https://doi.org/10.28919/ejma.2026.6.3