Local and global properties of solutions of quasilinear elliptic equations (Q1113004)

The main purpose of this paper is to study the global behavior and the isolated singularities of the positive radial solutions of the equation \[ \mathop{div}(| \nabla u|^{p-2}\nabla u(x))+u^ q(x)=0. \tag{E} \] The authors establish the following results. Let \(u\in C^ 1(\Omega)\setminus \{0\})\) be a positive radial solution of (E) in \(\Omega\setminus \{0\}\) then (i) either \(u\) can be extended to \(\Omega\) as \(C^ 1\) solution of (E) in \(\Omega\) or (ii) there exists an \(\alpha >0\) such that \[ \lim_{x\to 0} u(x)| x|^{(N-p)/(p-1)}=\alpha \quad\text{ if }p-1<q<N(p-1)/(N-p) \] or \[ \lim_{x\to 0} | x|^{p/(q+1-p)}u(x)=[(p/q+1-p)^{p-1}(N-pq/(q+1-p))]^{1/(q+1-1)}\tag{iii} \] \[ \text{ if } N(p-1)/(N- p)<q<Np/(N-p)-1 \] or \[ \lim_{x\to 0}[| x| (\log(1/| x|)] u(x)=[(N-p/p)((N-p)/(p-1))^{p-1}]^{(N-p)/(p-1)}\tag{iv} \] \[ \text{ if } q=N(p-1)/(N-p). \]