Mild integrability conditions for global solutions of an elliptic equation (Q1185972)

The author considers the semilinear equation (1) \(\Delta u+p(| x|)u^ \gamma=0\), \(x\in R^ n\) with the \(n\)-dimensional Laplacian operator for \(n\geq 3\). Assuming the continuity of \(p\) on \((a,\infty)\) for some \(a\geq 0\), \(\gamma\) be a real number \(\neq 0\) or 1, it is shown that the equation (1) has positive radially symmetric solutions on a given exterior domain \(E_ a=\{x\in R^ n;| x|>a\geq 0\}\). Furthermore, the author found out that the asymptotic behaviour of such solution is like constant multiples of the radial solutions \(v_ 1=1\) and \(v_ 2=| x|^{n-2}\) of \(\Delta v=0\). To prove these results the author derives several estimates for improper integrals. The integrability conditions on \(p\) are weaker than those usually imposed and the obtained estimates of the differences \(u-c\) or \(| x|^{n-2}u-c\) as \(| x|\to\infty\) are sharper than the known results in this subject.