Abstract

In Part 2, if \(B\in C^{N+1}(\mathbb R_+)\), \(B_1\in C^N(\mathbb R_+)\), \(B\geq b_0>0\), \(B_1\geq 0\), \(f\in C^{N+1}([0,1]\times\mathbb R)\) and \(f_1\in C^N([0,1]\times\mathbb R)\), we obtain from the following equation \(u_{tt}- [B(\|u_r\|_0^2)+\varepsilon B_1 (\|u_r\|_0^2)] (u_{rr}+ \frac1r u_r)= f(r,u)+\varepsilon f_1(r,u)\) associated to \((1)_{2,3}\) a weak solution \(u_\varepsilon(r,t)\) having an asymptotic expansion of order \(N+1\) in \(\varepsilon\), for \(\varepsilon\) sufficiently small.