M.Sc Thesis


M.Sc StudentHou Yongjun
SubjectOn Positive Solutions of the A-Laplacian with a Potential
DepartmentDepartment of Mathematics
Supervisors PROFESSOR EMERITUS Yehuda Pinchover
PROF. Antti Hermann Rasila
Full Thesis textFull thesis text - English Version


Abstract

In this thesis we study positive solutions of a quasilinear elliptic equation

in a subdomain ?? of Rn which is a generalization of the (p,A)-Laplace equation with a potential term V, where n??2,  1<p <??, the divergence of A is the well known A-Laplacian considered in the influential book of  Heinonen, Kilpeläinen, and Martio,

 and the potential V belongs to a certain local Morrey space. To the quasilinear  operator we naturally associate an energy functional defined on the space of  smooth functions with a compact support.


In addition to the structural assumptions on A as in the aforementioned book, we sometimes assume that A is locally strongly convex. Besides the prototype (p,A)-Laplacian, we provide operators fulfilling the aforementioned assumptions.


Based on the local Harnack inequalities, standard elliptic estimates, and the Harnack convergence principle, a systematic potential theory, called the criticality theory, is extended to the quasilinear operator .


Specifically speaking, employing suitable test functions, we prove an Agmon-Allegretto-Piepenbrink (AAP) type theorem, asserting that the nonnegativity of the associated functional is equivalent to the existence of a positive solution or positive supersolution of the quasilinear equation in ??. By virtue of a Picone-type identity and a generalized Hӧlder inequality, we present two alternative proofs of the AAP type theorem. The AAP type theorem and Picone's identity, respectively, imply the uniqueness and simplicity of a principal eigenvalue in a subdomain ?? of ?? . Furthermore, we establish the corresponding criticality theory.  Namely, nonnegative functionals are classified into  two classes: critical ones and subcritical ones.  A critical functional always admits a unique ground state, i.e., the limit of any null-sequence.We show that the criticality is equivalent to the nonexistence of a Hardy-weight. In addition, we extend to our setting a Hardy–Sobolev–Maz??ya inequality and a Liouville comparison theorem proved previously for the (p,A)-Laplacian with a potential. We also characterize the criticality in terms of the so-called (A,V)-capacity. Furthermore, by virtue of a weak comparison principle, which is derived from the super/sub-solution technique, we also study positive solutions of the equation of minimal growth at infinity in ??. In particular,  for every x0 in ??, there exists a positive solution in ?? \{x0}of minimal growth at infinity in ?? . Moreover, the criticality is also equivalent to the existence of a global minimal positive solution in ??. In fact, a  ground state is a global minimal positive solution in ??.