Multiple radial solutions for Dirichlet problem involving two mean curvature equations in Euclidean and Minkowski spaces

Yaning Wang

doi:10.1186/s13661-019-1199-8

Abstract

In this paper, we establish the existence of multiple positive radial solutions for the Dirichlet problem involving two mean curvature equations of spacelike graphs in Euclidean and Minkowski spaces.

Highlights

1 Introduction A graph Σu = {(x, u(x)) : x ∈ Ω} in the Euclidean space RN+1 is determined by a smooth function u(x) : Ω ⊆ RN → R
R3 with vanishing mean curvatures are necessarily the affine planes. Such a result holds for all 2 ≤ N ≤ 7 but is no longer true for higher dimension
Proof In what follows, we aim to prove that Lemma 2.2 is applicable under the above assumptions

Summary

Introduction

From a geometric point of view, the function f measures the proximity of the two different mean curvatures HL and HR of a spacelike graph in Minkowski and Euclidean spaces. Lemma 2.2 Let P be a cone in a real Banach space E, and let A : Pc → Pc be completely continuous and α be a nonnegative continuous concave functional on P with α(x) ≤ x for any x ∈ Pc. Suppose that there exist 0 < a < b < c such that the following conditions hold:. Since the proof for this step is very similar to that of Step 1, with the aid of condition (C1), we omit it

One can check easily that v

For any v