Radial and bifurcating non-radial solutions for a singular perturbation problem in the case of exchange of stabilities

We consider the singular perturbation problem −ε2Δu+(u−a(|x|))(u−b(|x|))=0 in the unit ball of RN, N⩾1, under Neumann boundary conditions. The assumption that a(r)−b(r) changes sign in (0,1), known as the case of exchange of stabilities, is the main source of difficulty. More precisely, under the assumption that a−b has one simple zero in (0,1), we prove the existence of two radial solutions u+ and u− that converge uniformly to max{a,b}, as ε→0. The solution u+ is asymptotically stable, whereas u− has Morse index one, in the radial class. If N⩾2, we prove that the Morse index of u−, in the general class, is asymptotically given by as ε→0, with c>0 a certain positive constant. Furthermore, we prove the existence of a decreasing sequence of εk>0, with εk→0 as k→+∞, such that non-radial solutions bifurcate from the unstable branch at ε=εk, k=1,2,…. Our approach is perturbative, based on the existence and non-degeneracy of solutions of a “limit” problem. Moreover, our method of proof can be generalized to treat, in a unified manner, problems of the same nature where the singular limit is continuous but non-smooth.