Symmetry and related properties via the maximum principle

Abstract

We prove symmetry, and some related properties, of positive solutions of second order elliptic equations. Our methods employ various forms of the maximum principle, and a device of moving parallel planes to a critical position, and then showing that the solution is symmetric about the limiting plane. We treat solutions in bounded domains and in the entire space.

Convergence for a Liouville equation

“…Taking a sequence λ n → 8mπ , we study the convergence of (u n ) , where This proposition actually can be derived from the works of Gidas, Ni, and Nirenberg [10] and De Figueiredo, Lions, and Nussbaum [28] (see [8] and [7] for related results). However, since it will play a very important role in the next section, we give a proof of it.…”

“…The idea of using the Pohozaev identities in asymptotic analysis has been known for many years and one may see [20] for related works cited there. As a side-remark, we point out that following [10] and [28], Chen and Li ([7] and [8]) used the moving plane method to obtain some beautiful apriori estimates for a class of elliptic partial differential equations in 2-dimensional domains. We remark that the special and important case when K = 1 was treated by K.Nagasaki and T.Suzuki (see [23] and [19]) using another method, which may also be applied to the case where K = constant (see [27]).…”

Resonance Phenomena in a Singular Perturbation Problem in the Case of Exchange of Stabilities

“…Moreover, there exists a constant C > 0 such that 20) and u ε → max{a, b} in C 0,α (Ω) as ε → 0, satisfying (1.15), for every 0 < α < 1 but not for α = 1.…”

“…This last fact also implies that w decays to zero super-exponentially as x → ±∞. Hence, since V + is even, and increasing for x > 0, we can apply the method of moving planes [20] to show that w is even and strictly decreasing for x > 0. Letting V ≡ V + − w, it is clear that V solves (3.20), is even, and strictly increasing for x > 0.…”

“…Firstly, we consider the following projected problem in H 2 (S): given e satisfying bound (5.18), 20) (recall that w = w andẼ = −E 1 for |x| ≤ 30γε − 1 3 ). We will prove that this problem has a unique solution whose norm is controlled by the L 2 -norm of E 12 , and not by that of the whole E 1 (recall (5.25)).…”