On positive solutions of the $(p,A)$-Laplacian with potential in Morrey space

Abstract

We study qualitative positivity properties of quasilinear equations of the form

$$Q_{A,p,V}^{\prime }\left[v\right]:=-div\left(|\nabla v{|}_A^{p-2}A\left(x\right)\nabla v\right)+V\left(x\right)|v{|}^{p-2}v=0,x\in \Omega ,$$

where $\Omega$ is a domain in ${ℝ}^n$, $1<p<\infty$, $A=\left(a_{ij}\right)\in L_{loc}^{\infty }\left(\Omega ;{ℝ}^{n\times n}\right)$ is a symmetric and locally uniformly positive definite matrix, $V$ is a real potential in a certain local Morrey space (depending on $p$), and

$$|\xi {|}_A^2:=A\left(x\right)\xi \cdot \xi =\sum _{i,j=1}^n{a}_{ij}\left(x\right){\xi }_i{\xi }_j,x\in \Omega ,\xi =\left({\xi }_1,\dots ,{\xi }_n\right)\in {ℝ}^n.$$

Our assumptions on the coefficients of the operator for $p\ge 2$ are the minimal (in the Morrey scale) that ensure the validity of the local Harnack inequality and hence the Hölder continuity of the solutions. For some of the results of the paper we need slightly stronger assumptions when $p<2$.

We prove an Allegretto–Piepenbrink-type theorem for the operator $Q_{A,p,V}^{\prime }$, and extend criticality theory to our setting. Moreover, we establish a Liouville-type theorem and obtain some perturbation results. Also, in the case $1<p\le n$, we examine the behaviour of a positive solution near a nonremovable isolated singularity and characterize the existence of the positive minimal Green function for the operator $Q_{A,p,V}^{\prime }\left[u\right]$ in $\Omega$.