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Soliton solutions for quasilinear Schrödinger equation with critical exponential growth in ℝN

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Abstract

In this work, we study the existence of nonnegative and nontrivial solutions for the quasilinear Schrödinger equation

$$- {\Delta _N}u + b{\left| u \right|^{N - 2}}u - {\Delta _N}\left( {{u^2}} \right)u = h\left( u \right)$$

, x ∈ RN where Δ N is the N-Laplacian operator, h(u) is continuous and behaves as exp(α|u|N/(N-1)) when |u| → ∞. Using the Nehari manifold method and the Schwarz symmetrization with some special techniques, the existence of a nonnegative and nontrivial solution u(x) ∈ W 1,N(RN) with u(x) → 0 as |x| → ∞ is established.

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Authors and Affiliations

  1. College of Science, Hohai University, Nanjing, 210098, P.R. China

    Caisheng Chen & Hongxue Song

  2. College of Science, Nanjing University of Posts and Telecommunications, Nanjing, 210023, P.R. China

    Hongxue Song

Authors
  1. Caisheng Chen
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  2. Hongxue Song
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Correspondence to Caisheng Chen.

Additional information

The research has been supported by the Fundamental Research Funds for the Central Universities of China (2015B31014) and NSFC-11571092.

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Chen, C., Song, H. Soliton solutions for quasilinear Schrödinger equation with critical exponential growth in ℝN . Appl Math 61, 317–337 (2016). https://doi.org/10.1007/s10492-016-0134-x

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  • DOI: https://doi.org/10.1007/s10492-016-0134-x

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