\[ \begin{array} { r l r } { ( 4 . 7 ) } & { \displaystyle \int _ { \sqrt { x _ { 1 } ^ { 2 } + x _ { n } ^ { 2 } } = \delta } \frac { 1 } { 2 } | \nabla u | ^ { 2 } ( \psi \cdot \nu ) | x _ { n } | ^ { 1 - 2 s } - \int _ { \sqrt { x _ { 1 } ^ { 2 } + x _ { n } ^ { 2 } } = \delta } ( \nabla u \cdot \psi ) ( \nabla u \cdot \nu ) | x _ { n } | ^ { 1 - 2 s } } \\ & { \displaystyle } & { \displaystyle + \int _ { B _ { 1 } ^ { \prime } } 2 \chi \operatorname { d i v } \varphi = O ( \delta ) . } \end{array} \]

where \( \nu \) is the unit outer normal to \( \left\{ { \sqrt { x _ { 1 } ^ { 2 } + x _ { n } ^ { 2 } } } = \delta \right\} \). Next we estimate the first two integrals in the LHS of the above equality. Since u is homogeneous of degree s (and depends only on \( x _ { 1 } \) and \( x _ { n } \)), \( \nabla \boldsymbol { u } \cdot \boldsymbol { \nu } = s \boldsymbol { u } \). Thus,

\[ \int _ { \sqrt { x _ { 1 } ^ { 2 } + x _ { n } ^ { 2 } } = \delta } ( \nabla u \cdot \psi ) ( \nabla u \cdot \nu ) | x _ { n } | ^ { 1 - 2 s } = \int _ { \sqrt { x _ { 1 } ^ { 2 } + x _ { n } ^ { 2 } } = \delta } ( \nabla u \cdot \psi ) ( s u ) | x _ { n } | ^ { 1 - 2 s } = O ( \delta ) , \]

where we have used the growth estimate of u around the zero set as well as\|∇u\|\|x−2sn\|12 ∈ L2loc(Rn). To estimate the first integral we use the polar coordinates in \( ( x _ { 1 } , x _ { n } ) \)-plane: \( x _ { 1 } = r \cos ( \theta ) \), \( x _ { n } = r \sin ( \theta ) \). Then on \( \{ { \sqrt { x _ { 1 } ^ { 2 } + x _ { n } ^ { 2 } } } = \delta \} \),

\[ \begin{array} { l } { \displaystyle \frac { 1 } { 2 } | \nabla u | ^ { 2 } ( \psi \cdot \nu ) | x _ { n } | ^ { 1 - 2 s } } \\ { \displaystyle \qquad = \frac { s ^ { 2 } \alpha ^ { 2 } } { 2 \delta } ( \cos ( \theta / 2 ) ) ^ { - 1 + 2 s } ( | \sin ( \theta ) | ) ^ { 1 - 2 s } \left[ \cos ( \theta ) ( \psi \cdot e _ { 1 } ) + \sin ( \theta ) ( \psi \cdot e _ { n } ) \right] } \end{array} \]

Since \( \psi \cdot e _ { n } = 0 \) on \( B _ { 1 } ^ { \prime } \),

\[ \begin{array} { l } { \displaystyle \operatorname* { l i m } _ { \delta \to 0 } \int _ { \sqrt { x _ { 1 } ^ { 2 } + x _ { n } ^ { 2 } } = \delta } \frac { 1 } { 2 } | \nabla u | ^ { 2 } ( \psi \cdot \nu ) | x _ { n } | ^ { 1 - 2 s } d x = \displaystyle \operatorname* { l i m } _ { \delta \to 0 } \int _ { B _ { 1 } ^ { \prime \prime } } \frac { s ^ { 2 } \alpha ^ { 2 } } { 2 \delta } \cdot 2 \delta \times } \\ { \displaystyle \quad \times \int _ { 0 } ^ { \pi } ( \cos ( \theta / 2 ) ) ^ { - 1 + 2 s } ( \sin ( \theta ) ) ^ { 1 - 2 s } ( \cos ( \theta ) ) ^ { 2 } \psi _ { 1 } ( \delta \cos ( \theta ) , x ^ { \prime \prime } , \delta \sin ( \theta ) ) \ d \theta d x ^ { \prime \prime } } \\ { \displaystyle \quad = c _ { 0 } ( s ) \alpha ^ { 2 } \int _ { B _ { 1 } ^ { \prime \prime } } \psi _ { 1 } ( 0 , x ^ { \prime \prime } , 0 ) d x ^ { \prime \prime } , } \end{array} \]

where

\[ \begin{array} { l } { { c _ { 0 } ( s ) = s ^ { 2 } \displaystyle \int _ { 0 } ^ { \pi } ( \cos ( \theta / 2 ) ) ^ { - 1 + 2 s } ( \sin ( \theta ) ) ^ { 1 - 2 s } ( \cos ( \theta ) ) ^ { 2 } d \theta } } \\ { { \displaystyle = s ^ { 2 } 2 ^ { - 1 - 2 s } \frac { \sqrt { \pi } ( 7 + 4 s ( s - 2 ) ) \Gamma ( 1 - s ) } { \Gamma ( \frac { 7 } { 2 } - s ) } . } } \end{array} \]

Combining the above estimates and letting \( \delta \rightarrow \) 0, we have for all \( \varphi \in C _ { c } ^ { \infty } ( B _ { 1 } ^ { \prime } ; \mathbb { R } ^ { n - 1 } ) \)

\[ c _ { 0 } ( s ) \alpha ^ { 2 } \int _ { B _ { 1 } ^ { \prime \prime } } \psi _ { 1 } ( 0 , x ^ { \prime \prime } , 0 ) d x ^ { \prime \prime } + \int _ { B _ { 1 } ^ { \prime } } 2 \chi \, \mathrm { d i v } \, \varphi = 0 \]

Recalling \( \psi _ { 1 } = \varphi _ { 1 } \) on \( B _ { 1 } ^ { \prime } \), we have proved (4.3).

Step 2. We now show that \( \begin{array} { r } { \alpha = \sqrt { \frac { 2 M } { c _ { 0 } ( s ) } } } \end{array} \) and \( \chi = M \chi _ { \{ x _ { 1 } > 0 \} } \)\( \mathcal { H } ^ { n - 1 } \)-a.e. in \( \mathbb { R } ^ { n - 1 } \).

(a) \( \chi \equiv M \) in \( \{ x _ { 1 } > 0 \} \).

In fact, if \( y \in \{ x _ { 1 } > 0 \} \), then \( u ( y ^ { \prime } , 0 ) = \alpha ( y _ { 1 } ) ^ { s } > 0 \). Hence by the uniform Ho¨lder convergence of \( u _ { \varepsilon _ { j } } \) to u, we have \( u _ { \varepsilon _ { i } } \geq \varepsilon _ { j } \) in a neighborhood of \( ( y ^ { \prime } , 0 ) \)

for any \( j \geq j _ { 0 } \) for some \( j _ { 0 } = j _ { 0 } ( \alpha , y _ { 1 } ) \) large enough. Thus for \( j > j _ { 0 } \) and \( ( x ^ { \prime } , 0 ) \) in the neighborhood of \( ( y ^ { \prime } , 0 ) \) we have

\[ \mathcal { B } _ { \varepsilon _ { j } } ( u _ { \varepsilon _ { j } } ) ( x ^ { \prime } ) = \int _ { 0 } ^ { u _ { \varepsilon _ { j } } ( x ^ { \prime } ) / \varepsilon _ { j } } \beta ( s ) d s = M . \]

Letting \( j \rightarrow \infty \) and since \( y ^ { \prime } \) is arbitrary, we get \( \chi \equiv M \) in \( \{ x _ { 1 } > 0 \} \).

(b) \( \chi \equiv \) 0 in \( \{ x _ { 1 } < 0 \} \).

In fact, we can take any \( \varphi \) in (4.3) such that \( \mathrm { s u p p } \, \varphi _ { 1 } \subset \mathbb { R } ^ { n - 1 } \cap \lbrace x _ { 1 } < 0 \rbrace \). Then the LHS of (4.3) will vanish. This implies that \( \chi = c o n s t \) in \( \{ x _ { 1 } < 0 \} \). By Lemma 3.2, \( \chi \equiv M \) or \( \chi \equiv \) 0 in \( \{ x _ { 1 } < 0 \} \). If \( \chi \equiv M \) in \( \{ x _ { 1 } < 0 \} \), then \( \chi \equiv M \) in \( \mathbb { R } ^ { n - 1 } \) by (a). Thus from (4.3) \( \begin{array} { r } { \int _ { \mathbb { R } ^ { n - 2 } } \varphi _ { 1 } ( 0 , x ^ { \prime \prime } ) d x ^ { \prime \prime } \, = \, 0 } \end{array} \) for any compactly supported vector field \( \varphi \), which is a contradiction.

This then implies the claim that \( \chi = M \chi _ { \{ x _ { 1 } > 0 \} } \). Next, applying an integration by parts to the RHS of (4.3) we have

\[ ( 4 . 8 ) \qquad \qquad \, \, c _ { 0 } ( s ) \alpha ^ { 2 } \int _ { B _ { 1 } ^ { \prime \prime } } \varphi _ { 1 } ( 0 , x ^ { \prime \prime } ) d x ^ { \prime \prime } = 2 M \int _ { B _ { 1 } ^ { \prime \prime } } \varphi _ { 1 } ( 0 , x ^ { \prime \prime } ) d x ^ { \prime \prime } . \]

This implies

\[ \alpha = \sqrt { \frac { 2 M } { c _ { 0 } ( s ) } } . \eqno \square \]

We are now ready to prove Theorem 4.1.

Proof of Theorem 4.1. Without loss of generality we assume \( x _ { 0 } \, = \, 0 \) and \( \nu = e _ { 1 } \). We also extend u by the even reflection with respect to \( x _ { n } \).

Consider the rescalings \( u _ { \lambda } \) and \( \chi _ { \lambda } \) at the origin. By Theorem 2.1 given \( \rho > \) 0,\( u _ { \lambda } \) is uniformly bounded in \( C ^ { 0 , s } ( \overline { { B _ { \rho / \lambda } } } ) \). Therefore, there exists a sequence \( \lambda _ { j } \to 0 \),\( u _ { 0 } \in C ^ { 0 , s } ( \mathbb { R } ^ { n } ) \) and \( \chi _ { 0 } \in L ^ { \infty } ( \mathbb { R } ^ { n } ) \) such that \( u _ { \lambda _ { j } } \to u _ { 0 } \) uniformly on compact subsets of \( \mathbb { R } ^ { n } \) and \( \chi _ { \lambda _ { j } } \rightharpoonup _ { \lambda 0 } ^ { * } \) in \( L ^ { \infty } ( \mathbb { R } ^ { n - 1 } ) \).

Now, rescaling (4.1), we see that for every \( R > \) 0,

\[ | \{ u _ { \lambda } ( \cdot , 0 ) > 0 \} \cap \{ x _ { 1 } < 0 \} \cap B _ { R } ^ { \prime } | \to 0 \quad { \mathrm { a s ~ } } \lambda \to 0 , \]

and we deduce that \( u _ { 0 } = 0 \) \( \mathcal { H } ^ { n - 1 } \)-a.e. in \( \{ x _ { 1 } < 0 , x _ { n } = 0 \} \). By continuity, \( \boldsymbol { u } _ { 0 } \) vanishes on all of \( \{ x _ { 1 } \, \leq \, 0 , x _ { n } \, = \, 0 \} \). Besides, we readily have that \( u _ { 0 } \) satisfies \( \mathrm { d i v } ( | x _ { n } | ^ { 1 - 2 s } \nabla u _ { 0 } ) = 0 \) in \( \{ u _ { 0 } > 0 \} \). Thus, we can apply Corollary A.2 in Appendix to obtain an asymptotic development

\[ ( 4 . 9 ) \qquad \qquad \qquad u _ { 0 } ( x ) = \alpha 2 ^ { - s } \left( ( x _ { 1 } ^ { 2 } + x _ { n } ^ { 2 } ) ^ { 1 / 2 } + x _ { 1 } \right) ^ { s } + o ( | x | ^ { s } ) \]

with \( \alpha \geq \) 0. Next, note that by rescaling the nondegeneracy condition (4.2) and passing to the limit, we have

\[ { \frac { 1 } { r ^ { n - 1 } } } \int _ { B _ { r } ^ { \prime } } u _ { 0 } d x ^ { \prime } \geq c \, r ^ { s } , \quad { \mathrm { f o r ~ a n y ~ } } r > 0 , \]

which implies that \( \alpha > \) 0. On the other hand, by Corollary 3.6(iii), \( u _ { 0 } \) is homo-geneous of degree s and hence

\[ u _ { 0 } ( x ) = \alpha 2 ^ { - s } \left( ( x _ { 1 } ^ { 2 } + x _ { n } ^ { 2 } ) ^ { 1 / 2 } + x _ { 1 } \right) ^ { s } . \]