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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    1,258 Downloads   2,034 Views  
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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

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.

References

[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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