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

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