OJAppS  Vol.2 No.4 B , December 2012
Branches of solutions for an asymptotically linear elliptic problem on RN
Abstract: 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,+∞).
Cite this paper: nullWan, Y. (2012) Branches of solutions for an asymptotically linear elliptic problem on RN. Open Journal of Applied Sciences, 2, 187-194. doi: 10.4236/ojapps.2012.24B043.

[1]   T.Bartsch,A.Pankov and Z.Q.Wang, Nonlinear Sch?dinger equations with steep potential well, Commun. Contemp. Math., 3(2001),549-569.

[2]   Y.Ding and K.Tanaka, Multiplicity of positive solutions of a nonlinear Sch?dinger equation, Manuscripta Math., 112(2003),109-135.

[3]   D.G.DE Figueiredo and Y.Ding, Solutions of a non-linear Schr?dinger equation, Discrete Contin. Dynam. Systems, 8(2002),563-584.

[4]   D.Gilbarg and N.S.Trudinger, it Elliptic Partial Diffential Equations of Second Order, Second edition, Springer-Verlag,Berlin, 1983.

[5]   F.A.Van Heerden and Z.Q.Wang, Schr?dinger tyoe equations with asymptotically linear nonlinearities, it Differential Integral Equations, 16(2003),257-280.

[6]   L.Jeanjean, M.Lucia and C.A.Stuart, Branches of solutions to semilinear ellptic equations on R^N, Math. Z., 230 (1999), 79-105.

[7]   L.Jeanjean and K.Tanaka, A positive solution for an asymptotically linear elliptic problem onR^N autonomous at infinity, WSAIM Control Optim. Calc. Var., 7(2002), 597-614.

[8]   Z. Liu and Z.Q. Wang, Existence of a positive solution of an elliptic equation onR^N, Proc. Roy. Soc. Edinburgh, 134 A (2004), 191-200.

[9]   C.A.Stuart, An introduction to elliptic equations on R^N, in Nonlinear Functional Analysis and Applications toDifferential Equations, editors A.Ambrosetti, K.C. Chang, I.Ekeland, World Scientific, Singapore,1998.

[10]   C.A.Stuart and Huansong Zhou, Global branch of solutios for nonlinear Schr?dinger equations with deepening potential well, Proc.London Math.Soc., 92 (2006) 655-681.

[11]   G.T. Whyburn, Topological Analysis, Princeton University Press, Preceton 1958.