an:04004735
Zbl 0619.35043
Galaktionov, V. A.; Kurdyumov, S. P.; Posashkov, S. A.; Samarskij, A. A.
A nonlinear elliptic problem with a complex spectrum of solutions
EN
U.S.S.R. Comput. Math. Math. Phys. 26, No. 2, 48-54 (1986); translation from Zh. Vychisl. Mat. Mat. Fiz. 26, No. 3, 398-407 (1986).
00166070
1986
j
35J60 35K55
radial-symmetric positive solution; nonlinear elliptic; automodel solutions; quasilinear parabolic; discrete (denumerable) solutions; continual solution; bifurcation
The authors consider the radial-symmetric positive solution of the nonlinear elliptic equation in \({\mathbb{R}}^ N\) arising from the study of the unbounded automodel solutions of the quasilinear parabolic equation with the source
\[
u_ t=\nabla (| \nabla u|^{\sigma}\nabla u)+u^{\beta},\quad t>0,\quad x\in {\mathbb{R}}^ N,\quad \sigma,\beta \in R,\quad \sigma >0,\quad \beta >\sigma +1.
\]
It is shown that the elliptic problem has 4 different families of solutions; three discrete (denumerable) solutions and one continual solution. In the one- dimensional case the solutions are constructed numerically, the bifurcation situation is given.
Ya.A.Rojtberg