Take a change z = r$_{+}$ / r , Eqs. (3.4) and (3.5) can be rewritten as where the prime denotes di ff erentiation with z . Regularity of the functions at the horizon z = 1 requires And near the boundary z = 0 we have We will set $^{C}$− = 0 and fix ρ in the following discussion. With the help of the regular horizon boundary condition (3.13), the leading order approximate solutions near the horizon, z = 1, for the Eqs. (3.11) and (3.12) can be expressed as where a ≡ ψ (1) and b ≡ − φ ′ (1) with a , b > 0 which makes ψ ( z ) and φ ( z ) positive near the horizon. Matching smoothly the solutions (3.15), (3.16) with (3.14) at an intermediate point z$_{m}$ with 0 < z$_{m}$ < 1, we have with 8 A = 4 L $^{4}$m$^{4}$(1 − z$_{m}$ )[ λ$_{+}$ − ( λ$_{+}$ − 2) z$_{m}$ ] + 36 λ$_{+}$ + 3 L $^{2}$m$^{2}$[(1 + 3 ϵ $^{2}$)( λ$_{+}$ − 2) z 2 m − 2(5 + 3 ϵ $^{2}$)( λ$_{+}$ − 1) z$_{m}$ + 3(3 + ϵ $^{2}$) λ$_{+}$ ] . (3.20) a 2 = 3 4 L $^{2}$(1 − z$_{m}$ ) ( ρ br$_{+}$ ) ( 1 − br$_{+}$ ρ ) , (3.19) b = r$_{+}$ 2 L 2 √ A [ λ$_{+}$ − ( λ$_{+}$ − 2) z$_{m}$ ](1 − z$_{m}$ ) ≡ ˜ br$_{+}$ L 2 , (3.18) C$_{+}$ = 6 + 2 m $^{2}$L$^{2}$(1 − z$_{m}$ ) 3[2 z$_{m}$ + λ$_{+}$ (1 − z$_{m}$ )] z λ$_{+}$ − 1 m a , (3.17) φ ( z ) = b [ (1 − z ) + 2 L $^{2}$a2 3 (1 − z ) 2 + · · · ] , (3.16) ψ ( z ) = a { 1 + 2 m $^{2}$L2 3 (1 − z ) + L 2 36 [ (3 + 9 ϵ $^{2}$+ 4 m $^{2}$L$^{2}$) m 2 − 4 L $^{2}$b2 r 2 + ] (1 − z ) 2 + · · · } , (3.15) ψ = $^{C}$−z $^{λ}$− + C$_{+}$z λ$_{+}$ , φ = µ − ρ r$_{+}$ z . (3.14) ψ (1) = 3 2 m $^{2}$L$_{2}$ψ ′ (1) , φ (1) = 0 . (3.13) φ ′′ − r 2 + z 4 2 ψ 2 f φ = 0 , (3.12) ψ ′′ + f ′ f ψ ′ + r 2 + z 4 ( φ 2 f 2 − m 2 f ) ψ = 0 , (3.11)