On the positive solutions for \(\triangle_mU+g(|x|,U,|\nabla U|)=0\) in \(\mathbb{R}^n\) (Q2706034)

The author investigates the existence of solutions for the problem \(D_m U+r^ag(r,U,|U'|)=0 \), \(U'(0)=0\) \((U>0)\), where \(U\in C_m^1\equiv \{\Phi\in C^1(\mathbb R_+); \;r^a|\Phi '|^{m-2}\Phi '\in C^1(\mathbb R_+)\}\), \(D_mU\equiv \{ r^a|U'|^{m-2}U'\} '\), \(r\equiv |x|\), \(\{\cdot \}'\equiv (d/dr)\{\cdot \}\). It is considered a more general function \(g\) by including the hypothesis \(v'\leq u'\) in the definition of the sub-super-solutions \(u\) and \(v\). This is motivated by the fact that by using \(v(r)=(1+r^s)^{-b}\) and \(u(r)=(A+r^k)^{-b}\) with \(s,A>1\), \(b>0\) and \(k>2\) for the sub-super-solutions, respectively, the inequality \(v'<u'\) easily holds when \(g(r,t,T)\) is non decreasing in \(t,T>0\).