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We consider  the following nonlinear schr?dinger equation -?u+λV(x)u=f(x,u)withu∈H^1 (R^N) and u?0,(*) whereλ>0 and f(x,s) is asymptotically linear withrespect to sat origin and infinity. The potential V(x) satisfies V(x)≥V_0>0for all x∈R^N and (_|x|→+∞^lim)V(x)=V (∞)∈(0,+∞). We provethat problem (*) has two connected sets of positive and negative  solutions inR×W^(2,p) (R^N)for somep∈[2,+∞)∩(N/2,+∞).

Branches of solutions for an asymptotically linear elliptic problem on R<sup>N</sup>