Pointwise estimates for systems of coupled $ p $-Laplacian elliptic equations

Abstract

This work examines positive solutions of systems of inequalities$ \pm \Delta_{{\bf{p}}} {\bf{u}} \ge \rho(x) {\bf{f}}({\bf{u}}), \qquad \mbox{ in } \Omega, $where $ {\bf{p}} = (p_1,...,p_k) $, $ p_i > 1 $ and $ \Delta_{{\bf{p}}} $ is the diagonal-matrix $ \text{diag}(\Delta_{p_1},...,\Delta_{p_k}) $, $ \Delta_{p_i} $ is the $ p_i $-Laplace operator, $ \Omega $ is an arbitrary domain (bounded or not) in $ {\mathbb{R}}^N $ ($ N\ge 2 $), $ {\bf{u}} = (u_1,...,u_k)^T $ and $ {\bf{f}} = (f_1,...,f_k)^T $ are vector-valued functions and $ \rho(x) $ is a nonnegative function in $ \Omega $ which is locally bounded. Using a maximum principle-based argument we provide explicit estimates on positive solutions $ u $ at each point $ x\in\Omega $, and as applications we find Liouville type results in unbounded domains such as $ {\mathbb{R}}^N $, exterior domains or generally unbounded domains with the property that $ \sup_{x\in\Omega}dist (x,\partial\Omega) = \infty $, for various nonlinearities $ {\bf{f}} $ and weights $ \rho $. We also give explicit upper bounds on extremal parameters of related nonlinear multi-parameter eigenvalue problems in bounded domains.