On the equation \(\Delta u+x\cdot \nabla u+f(u)=0\)
The asymptotic behaviour of radial solutions of the equation \[ (1)\quad \Delta u+\frac{1}{2}x\cdot \nabla u+\frac{k}{2}u+| u|^{p- 1}u=0,\quad x\in R^ N, \] which is an ordinary differential equation \[ (2)\quad u''+(\frac{N-1}{r}+\frac{r}{2})u'+\frac{k}{2}u+| u|^{p-1}u=0,\quad r=| x|,\quad u(0)=a,\quad u'(0)=0 \] for such solutions is investigated. It is proved that if \(\lim_{r\to \infty}r^ k u(r)=0\), then \(u(r)=0(e^{-\frac{r^ 2}{4}})\) and if \(\lim_{r\to \infty}r^ k u(r)\neq 0\), then \(u(r)=0(r^{-k})\). In this connection two asymptotic terms depending on the symbol of the variable (p-1)k-2 are calculated and it is shown that for \(k\leq \frac{N}{2}\), \((p+1)(p-1)^{-1} \leq \frac{N}{2}\), \(u(r)>0\) for \(r>0\) and \(\lim_{r\to \infty}r^ k u(r)>0\). For the solution of equation (1) and \[ \Delta u- \frac{1}{2}x\cdot \nabla u-\frac{k}{2}u+| u|^{p-1} u=0 \] integral identities generalizing the Pokhozhaev formula were proved [\textit{S. I. Pokhozhaev}, Dokl. Akad. Nauk SSSR 165, 36-39 (1965; Zbl 0141.302)]. The explicit formulae of solutions of equation (2) for certain combinations between the parameters N, k and p are given at the end.
scientific article; zbMATH DE number 3996226
scientific article