Abstract :
We consider the sub- or supercritical Neumann elliptic problem −Δu+μu=u5+ε, u>0 in Ω; ∂u∂n=0 on ∂Ω, Ω being a smooth bounded domain in R3, μ>0 and ε≠0 a small number. Hμ denoting the regular part of the Greenʹs function of the operator −Δ+μ in Ω with Neumann boundary conditions, and ϕμ(x)=μ12+Hμ(x,x), we show that a nontrivial relative homology between the level sets ϕμc and ϕμb, b0 small enough, of a solution to the problem, which blows up as ε goes to zero at a point a∈Ω such that b⩽ϕμ(a)⩽c. The same result holds, for ε<0, assuming that 00) for μ small (resp. large) enough, providing us with cases where the above assumptions are satisfied.
