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Nonstandard solutions for a perturbed nonlinear Schrödinger system with small coupling coefficients

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  • Xiaoming An
  • Chunhua Wang

Abstract

In this paper, we consider the following weakly coupled nonlinear Schrödinger system −ε2Δu1+V1(x)u1=|u1|2p−2u1+β|u1|p−2|u2|pu1,x∈RN,−ε2Δu2+V2(x)u2=|u2|2p−2u2+β|u2|p−2|u1|pu2,x∈RN,where ε>0, β∈R is a coupling constant, 2p∈(2,2∗) with 2∗=2NN−2 if N≥3 and +∞ if N=1,2, V1 and V2 belong to C(RN,[0,∞)). When p≥2 and β>0 is suitably small, we show that the problem has a family of nonstandard solutions {wε=(uε1,uε2):0 0 is close to 0 in this paper. Moreover, the location of concentration points is given by local Pohozaev identities. Our proofs are based on variational methods and the penalized technique.

Suggested Citation

  • Xiaoming An & Chunhua Wang, 2021. "Nonstandard solutions for a perturbed nonlinear Schrödinger system with small coupling coefficients," Mathematische Nachrichten, Wiley Blackwell, vol. 294(6), pages 1052-1084, June.
    • Handle: RePEc:bla:mathna:v:294:y:2021:i:6:p:1052-1084
    DOI: 10.1002/mana.202000121
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