Stability Results of Positive Weak Solution for a Class of Chemically Reacting Systems

Abstract

This paper aims to study the existence and non-existence results of positive weak solution to the quasilinear elliptic system:
\[
\begin{cases}
-\Delta_p u = \lambda a(x) \left[ f(u,v) - \dfrac{1}{u^\alpha} \right], & x \in \Omega, \\
-\Delta_q v = \lambda b(x) \left[ g(u,v) - \dfrac{1}{v^\beta} \right], & x \in \Omega, \\
u = 0 = v, & x \in \partial\Omega,
\end{cases}
\]
where \(\Delta_r w = \operatorname{div}(|\nabla w|^{r-2} \nabla w)\) is the \(r\)-Laplacian (\(r = p, q\)), \(r > 1\), \(\alpha, \beta \in (0,1)\), \(\Omega\) is a bounded domain in \(\mathbb{R}^N\) (\(N > 1\)) with smooth boundary \(\partial\Omega\) and \(\lambda\) is a positive parameter. Here \(f, g\) are \(C^1\) increasing functions such that \(f, g : \mathbb{R}^+ \times \mathbb{R}^+ \rightarrow \mathbb{R}^+\); \(f(\upsilon_1, \upsilon_2) > 0\), \(g(\upsilon_1, \upsilon_2) > 0\) for \(\upsilon_1, \upsilon_2 > 0\). With \(C^1\) sign-changing functions \(a(x)\), \(b(x)\) that perhaps have negative values nearby the boundary. We establish our results via the sub-supersolution method. In addition, we study the stability and instability results of positive weak solution with different choices of \(f\) and \(g\).