Bounded positive entire solutions of multi-dimensional Emden-Fowler type equations with oscillating coefficients (Q1174361)

The semilinear elliptic Emden-Fowler equation: \(\Delta u+a(| x|)f(u)=0\) in \(\mathbb{R}^ N\) is considered, where \(N\geq 3\), \(f(u)>0\) and \(f'(u)>0\) for \(u>0\), and \(f(u)\) satisfies either \(\lim_{u\to+0}f'(u)=\lim_{u\to+0}f(u)/u=0\) or \(\lim_{u\to\infty}f(u)=0\). Introducing conditions on \(a(t)\) in terms of \(A(t)=\int_ 0^ t (s/t)^{N-1}a(s)ds\) and \(\alpha(t)=\int_ t^ \infty a(s)ds\) and observing that \(a(t)\) may oscillate, the authors improve some known results about the existence of uniformly positive radial entire solutions and further derive the asymptotic properties of the solutions at \(| x|=\infty\). They also consider the nonradial equation \(\Delta u+b(x)f(u)=0\).