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Branches of solutions for an asymptotically linear elliptic problem on RN

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DOI: 10.4236/ojapps.2012.24B043    1,295 Downloads   2,086 Views  
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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,+∞).

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The authors declare no conflicts of interest.

Cite this paper

Wan, 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.

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