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On the equation $$\Delta u+x\cdot \nabla u+f(u)=0$$. (English) Zbl 0615.35034
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 [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.
Reviewer: B.Kvedaras

##### MSC:
 35J60 Nonlinear elliptic equations 35B40 Asymptotic behavior of solutions to PDEs 35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
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