Criticality theory of half-linear equations with the (p,A)-Laplacian

Abstract

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.