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 $$Q_{A,V}(u):= -\mathrm{div} (|\nabla u|_{A}^{p-2}A(x)\nabla u)+ V(x)|u|^{p-2}u=0 \quad \mbox{in }Ω,$$ where $1<p<\infty$, $\Omega$ is a domain in ${\mathbb{R}}^n$, $n\ge 2$, $V\in L_{\mathrm{loc}}^{\infty }(\Omega )$, $A=\left(a_{ij}\right)\in L_{\mathrm{loc}}^{\infty }(\Omega ,{\mathbb{R}}^{n^2})$ is a symmetric and locally uniformly positive definite matrix in $\Omega$, and $$|\xi {|}_A^2:=⟨A(x)\xi ,\xi ⟩=\sum _{i,j=1}^n{a}_{ij}(x){\xi }_i{\xi }_{j}x\in \Omega ,\xi =({\xi }_1,\dots ,{\xi }_n)\in {\mathbb{R}}^n.$$ 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.

arXiv comment29 pages