M.Sc Thesis | |

M.Sc Student | Hou Yongjun |
---|---|

Subject | On Positive Solutions of the A-Laplacian with a Potential |

Department | Department of Mathematics |

Supervisors | PROFESSOR EMERITUS Yehuda Pinchover |

PROF. Antti Hermann Rasila | |

Full Thesis text |

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

in a subdomain ??
of R^{n} 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 x_{0} _{ }in ??, there exists a positive solution in ?? \{x_{0}}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 ??.