Positive solutions to elliptic equations in unbounded cylinder

Abstract

This paper investigates the positive solutions for second order linear elliptic equation in unbounded cylinder with zero boundary condition. We prove there exist two special positive solutions with exponential growth at one end while exponential decay at the other, and all the positive solutions are linear combinations of these two.

Classification of positive solutions for fully nonlinear elliptic equations in unbounded cylinders

“…They proved that the space of the positive solutions in a cone with zero boundary value is one-dimensional. These results have been extended by Jun Bao and some of the authors [4] in unbounded cylinders. They showed that there are two special solutions with exponential growth at one end while exponential decay at the other and all the positive solutions are linear combinations of these two.…”

“…Proof of Theorem 1.4: From Lemma 6.2 and by using the similar arguments in the proof of Theorem 1.2 in [4], we can prove Theorem 1.4. Here we omit the details.…”

“…Following the original strategy of Jun Bao and some of the authors [4] as well as the two fundamental tools, this paper deals with the following fully nonlinear uniformly elliptic equations…”

Stability of Bernstein type theorem for the minimal surface equation

“…First we mention that (3.1)-(3.2) has been studied in special cases when the domain Ω is an infinite cone or an infinite cylinder (see [14] and [3] respectively). In both cases, it was proved that the dimension of the space consisting of all solutions is determined by the number of ends of the underlying domain Ω.…”

“…This philosophy, however, remains to be an open problem when Ω turns out to be a general unbounded domain. One cannot apply those methods from [14,3] to general cases. Indeed, their arguments rely heavily on the scaling or translating invariance of the underlying domain Ω, which guarantees that a Harnack inequality with a uniform constant even holds around the infinity.…”