Asymptotic transversality and symmetry breaking bifurcation from boundary concentrating solutions

Abstract

Let A:={a<|x|<1+a}⊂RN and p⩾2. We consider the Neumann problemε2Δu−u+up=0in A,∂νu=0on ∂A. Let λ=1/ε2. When λ is large, we prove the existence of a smooth curve {(λ,u(λ))} consisting of radially symmetric and radially decreasing solutions concentrating on {|x|=a}. Moreover, checking the transversality condition, we show that this curve has infinitely many symmetry breaking bifurcation points from which continua consisting of nonradially symmetric solutions emanate. If N=2, then the closure of each bifurcating continuum is locally homeomorphic to a disk. When the domain is a rectangle (0,1)×(0,a)⊂R2, we show that a curve consisting of one-dimensional solutions concentrating on {0}×[0,a] has infinitely many symmetry breaking bifurcation points. Extending this solution with even reflection, we obtain a new entire solution.