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Multiple soliton solutions for a quasilinear Schrödinger equation

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Abstract

Using Morse theory, truncation arguments and an abstract critical point theorem, we obtain the existence of at least three or infinitely many nontrivial solutions for the following quasilinear Schrödinger equation in a bounded smooth domain

$$\left\{ {\begin{array}{*{20}{c}} { - {\Delta _p}u - \frac{p}{{{2^{p - 1}}}}u{\Delta _p}\left( {{u^2}} \right) = f\left( {x,u} \right)\;in\;\Omega } \\ {u = 0\;on\;\partial \Omega .} \end{array}} \right.$$
((0.1))

Our main results can be viewed as a partial extension of the results of Zhang et al. in [28] and Zhou and Wu in [29] concerning the the existence of solutions to (0.1) in the case of p = 2 and a recent result of Liu and Zhao in [21] two solutions are obtained for problem 0.1.

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Authors and Affiliations

  1. School of Mathematics and Information Science, Beifang University of Nationalities, Yinchuan, 750021, China

    Jiayin Liu

  2. School of Mathematics and Statistics, Lanzhou University, Lanzhou, 730000, China

    Duchao Liu

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  1. Jiayin Liu
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  2. Duchao Liu
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Correspondence to Jiayin Liu.

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Liu, J., Liu, D. Multiple soliton solutions for a quasilinear Schrödinger equation. Indian J Pure Appl Math 48, 75–90 (2017). https://doi.org/10.1007/s13226-016-0195-2

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  • DOI: https://doi.org/10.1007/s13226-016-0195-2

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