Decaying entire positive solutions of quasilinear elliptic equations (Q1071946)

In this paper the quasilinear elliptic equation \[ (*)\quad \Delta u+f(x,u,\nabla u)=0,\quad x\in {\mathbb{R}}^ N,\quad N\geq 3, \] is considered, where \(\nabla =(\partial /\partial x_ 1,...,\partial /\partial x_ N)\), \(\Delta =\nabla \cdot \nabla\), and f is nonnegative and locally Hölder continuous in \({\mathbb{R}}^ N\times {\mathbb{R}}_+\times {\mathbb{R}}^ N\), \({\mathbb{R}}_+=(0,\infty).\) Under additional structure hypotheses it is shown that equation (*) has an entire positive solution which decays to zero at infinity. In particular, conditions are established for the existence of an entire positive solution of (*) which behaves like a constant multiple of \(| x|^{2-N}\) as \(| x| \to \infty\). Examples to which our existence theory is applicable are \[ \Delta u+p(x)u^{\gamma}+\sum^{m}_{i=1}q_ i(x) u^{\rho_ i} | \nabla u|^{\sigma_ i}=0,\quad \Delta u+p(x)u^{- \gamma}+\sum^{m}_{i=1}q_ i(x) u^{-\rho_ i} | \nabla u|^{\sigma_ i}=0, \] where \(\gamma\), \(\rho_ i\), \(\sigma_ i\) are nonnegative constants satisfying \(0\leq \gamma <1\), \(0<\rho_ i+\sigma_ i<1\), \(i=1,...,m\).