EXISTENCE AND ASYMPTOTIC BEHAVIOR OF SINGULAR SOLUTIONS OF QUASILINEAR ELLIPTIC EQUATIONS IN Rn

Abstract

In this paper we consider the quasilinear elliptic equation \def\theequation{1} \begin{equation} {\rm div}(|\nabla u|^{m-2}\nabla u)+f(u)=0 \end{equation} where $n\gt m\gt 1$. We obtain a necessary and sufficient condition for the existence of positive radial solutions $u=u(r)$ on $[r_0, \infty)$, where $r_0 \gt 0$. If $f$ satisfies a further condition, then Eq. (1) possesses infinitely many singular ground state solutions $u(r)$ satisfying $u(r)\sim r^{-{(n-m)}\over {m-1}}$ at $\infty $ and $u(r)\to \infty \hbox{\ as\ }r\to 0^+$. We also obtain some important conclusions via our main results.