We study positive solutions of half-linear second-order elliptic equations of the form $$Q_{ A,V}(u):= -\mathrm{ div} (|\nabla u|_{ A}^{ p-2}A(x)\nabla u)+ V(x)|u|^{ p-2}u=0 \quad \mbox{ in }\Omega,$$ where $1<p<\infty$, $\Omega$ is a domain in $\mathbb{ R}^{ n}$, $n\geq 2$, $V\in L_{ \mathrm{ loc}}^{ \infty}(\Omega)$, $A=\big(a_{ ij}\big)\in L_{ \mathrm{ loc}}^{ \infty}(\Omega,\mathbb{ R}^{ n^2})$ is a symmetric and locally uniformly positive definite matrix in $\Omega$, and $$|\xi|_{ A}^{ 2}:=\left\langle A(x)\xi,\xi\right\rangle=\sum_{ i,j=1}^n a_{ ij}(x)\xi_i\xi_j \qquad x\in \Omega, \xi=(\xi_1,\ldots,\xi_n)\in\mathbb{ R}^n.$$ We extend criticality theory which has been established for linear operators and for half-linear operators involving the $p$-Laplacian, to the operator $Q_{ A,V}$. We prove Liouville-type theorems, and study the behavior of positive solutions of the equation $Q_{ A,V}(u)=0$ near an isolated singularity and near infinity in $\Omega$, and obtain some perturbations results.
Categories: math.AP