We consider the stationary non-linear Schrödinger equation Δu+{ 1+λg(x) }u=f(u)with u∈H1RN,u ≠0 where λ > 0 and the functions f and g are such that lims→0f(s)s=0and1<α+1=lim| s |→∞f(s)s<∞d and g(x)≡0on Ω¯,g(x)∈(0,1]on RN\Ω¯and lim| x |→+∞g(x)=1 for some bounded open set Ω ∈ RN. We use topological methods to establish the existence of two connected sets D± of positive/negative solutions in R × W2, p RN where p∈[2,∞)∩(2N,∞) that cover the interval (α, Λ(α)) in the sense that PD±=(α,Λ(α))where P(λ,u)=λ, and furthermore, limλ→Λ(α)-‖ uλ ‖L∞(RN)=limλ→Λ(α)-‖ uλ ‖W2,p(RN)=∞,for (λ,uλ)∈D± The number Λ(α) is characterized as the unique value of λ in the interval (α, ∞) for which the asymptotic linearization has a positive eigenfunction. Our work uses a degree for Fredholm maps of index zero. 2000 Mathematics Subject Classification 35J60, 35B32, 58J5
