A comparison type principle for a class of quasilinear elliptic systems: critical and subcritical cases (Q2830488)

The paper deals with solutions of the system NEWLINE\[NEWLINE \begin{cases} -\Delta_{p(x)} u= u^{a(x)} v^{b(x)}, &\quad x\in\Omega,\\ -\Delta_{q(x)} v= u^{c(x)} v^{d(x)}, &\quad x\in\Omega,\\ u>0,\;v>0 ,\end{cases} NEWLINE\]NEWLINE with different Dirichlet boundary conditions such as \(u=\lambda\), \(v=\mu\), or \(u=v=+\infty\) or \(u=\lambda\), \(v=+\infty\) on \(\partial\Omega\). Here \(\Omega\subset\mathbb R^N\) is a bounded and smooth domain, \(\lambda,\mu>0\), \(p,q: \overline{\Omega}\to\mathbb R\) are continuous functions with \(p(x),q(x)>1\), \(a(x)>p(x)-1\), \(d(x)>q(x)-1\). On the base of a comparison type principle, the authors study existence, non-existence, uniqueness or multiplicity of positive solutions in both the critical and subcritical cases.