Some notes on the method of moving planes

Abstract

In this paper, we obtain a version of the sliding plane method of Gidas, Ni and Nirenberg which applies to domains with no smoothness condition on the boundary. The method obtains results on the symmetry of positive solutions of boundary value problems for nonlinear elliptic equations. We also show how our techniques apply to some problems on half spaces.In this note, we show how to apply the method of moving planes [6] Gidas and Spruck [8].In Section 1, we prove our main result on bounded domains and discuss generalisations and in Section 2 we prove some technical lemmas needed in Section 1. In Section 3, we consider the problem on half spaces.

Liouville-type theorems and decay estimates for solutions to higher order elliptic equations

“…The proof of their results is based on an idea of [9]. The idea is the following: if there exists a solution of (1.3) and one is able to show that any solution is increasing in the x N direction, then passing the limit as x N → ∞, one could get a solution of the same equation in R N−1 , which in turn allows the use of the Liouville-type theorem in the whole space.…”

“…(Note that the critical exponent in (1.3) is N−1+2m N−1−2m .) We would like to mention that the Liouville-type theorem for (1.3) in [9] is for the case of m = 1 with the assumption that the solution is bounded. Later on this result was improved to hold for unbounded solution in [2], among many other results.…”

“…Among many results in [15], the Liouville-type theorem for the subcritical elliptic equation is shown. Namely, there is no nontrivial C 2 solution of the problem 9) if N 3 and 1 < p < N+2 N−2 . See also [6] for a simpler proof with the Kelvin transform and moving plane method of Gidas, Ni and Nirenberg [14].…”

Existence and non-existence results for fully nonlinear elliptic systems

“…We use an idea by Dancer [13], which consists in the following : if there is a solution of the problem in {x N > 0}, and if one is able to show that any such solution is increasing in the x N -direction, then, after eventually some supplementary work, one should be able to pass at the limit as x N → ∞ and thus get a solution of the same problem in R N −1 , which in turn permits to use the nonexistence result for the whole space, that we already proved. Note that the numbers N 1 , N 2 from Theorem 1.3 (which we defined in the previous sections) are strictly increasing in N , so the nonexistence results in R N + hold for a larger range of p, q than the nonexistence result in R N .…”

Semiclassical analysis of general second order elliptic operators on bounded domains

“…The result is well known for the full space case; it can be proved for instance by obtaining a linear differential equation for the average of w on spheres. In the half space case one can show that ∂ j w > 0 provided the half-space is x j > 0 (by a variant of a result in [6]), and then show that lim xj→∞ w satisfies a full space equation in one lower dimension.…”