One bubble against a ring: new sign-changing solutions of the Brezis-Nirenberg problem

Abstract: We consider the Brezis--Nirenberg problem $-Δu=|u|^{{2^{*-2}u+\varepsilon}} u$ in the unit ball of $\mathbb R<sup>N$, $N\ge5$, with Dirichlet boundary conditions, and we look for sign-changing solutions which, as $\varepsilon\to0<sup>+$, blow up positively at the centre and negatively at the $k$ vertices of a regular polygon. The key observation is that, for this configuration, the gradient of the reduced energy factorises, so that the whole finite-dimensional problem is governed by the zeros of a single explicit function of the radius of the polygon. Its behaviour depends on the dimension: when $N=5$ such solutions exist for every $k\ge2$, and the ring of negative peaks approaches the boundary at distance $O(1/k)$; when $N\ge6$ there are two solutions for $2\le k\le6$, while no solution of this type exists for $k$ large. For $2\le k\le6$ the nodal set of every such solution reaches the boundary of the ball, in contrast with what happens for the almost critical problem. Nevertheless, the two solutions found for $N\ge6$ have very different nodal structures: numerically, for $3\le k\le6$ one of them has only two nodal domains, and its nodal set meets the plane of the peaks along a closed curve around the centre, while the other one can have as many as $k+1$ nodal domains. These results extend to higher dimensions a configuration previously found in dimension four.