Non-Radially Symmetric Solutions for Some Sublinear Defocusing Elliptic Problems

In this paper, we study the action of the orthogonal group O(n) on the set of solutions of some sub-linear elliptic problem of the form \(\Delta u\) + \(u\) − | \(u\) |−2\(\theta\) \(u\) = 0 on the unit ball B of \(\mathbb{R}\)n, n ≥ 2 and 0 < 2θ < 1. For suitable subgroups G of the orthogonal group O(n), we show the existence of non-radially symmetric solutions, which are G-invariant. We precisely give a necessary and sufficient condition on G for the existence of non-radially symmetric but G-invariant solutions. Besides, we develop complete proofs for the existence and uniqueness of radial solutions. Recall that a majority of studies of this type of problem were focusing on radial solutions which are obviously invariant by the group O(n). On the other hand, the question of the existence of non-radial solutions remaines somehow abandoned. In this paper, we studied this question and provided computer numerical simulations for radial solutions and the estimate of the Sobolev norm of the new non radial solutions.