On radially symmetric solutions of the equation −Δu + u = |u| p-1 u. An ODE approach

摘要

Questions of the existence in a ball of radially symmetric solutions of the equation indicated in the title with the Dirichlet zero boundary conditions are studied in many publications and generally speaking, there was obtained more or less complete answer on these questions. It is known now that if the dimension of the space d ≥ 3 and 1 p (d + 2)/(d − 2) or if d = 2 and p 1, then for any integer l ≥ 0 this problem in a ball or in the entire space ${x in mathbb {R}^d}$ has a radially symmetric solution with precisely l zeros as a function of r = |x|. If d ≥ 3 and p ≥ (d + 2)/(d − 2), then the problem in the entire space has no nontrivial solution. For the first time, this problem was studied by a variant of the variational method. However, it is known to the specialists in the field that it is also interesting to obtain the same results by using methods of the qualitative theory of ODEs. In the present article, we shall give a simple proof of the result above in this way. An earlier proof of this result of the other authors is essentially more complicated than our one.