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On a class of quasilinear Schrödinger equations with vanishing potentials and mixed nonlinearities

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Abstract

In this paper, we study the following generalized quasilinear Schrödinger equations with mixed nonlinearity

$$\left\{ {\begin{array}{*{20}{c}} { - div({g^2}(u)\nabla u) + g(u)g'(u){{\left| {\nabla u} \right|}^2} + V(x)u = K(x)f(u) + \lambda \xi (x)g(u){{\left| {G(u)} \right|}^{p - 2}}G(u), x \in {\mathbb{R}^N},} \\ {u \in {\mathcal{D}^{1,2}}({\mathbb{R}^N}),} \end{array}} \right.$$

where N ≥ 3, V, K are nonnegative continuous functions and f is a continuous function with a quasicritical growth. Using a change of variable as \(G(u) = \int_0^u {g(t)\rm{dt}}\), the above quasilinear equation is reduced to a semilinear one. Under some suitable assumptions, we prove that the above equation has at least one nontrivial solution by working in weighted Sobolev spaces and employing the variational methods.

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Authors and Affiliations

  1. School of Mathematics and Computational Science, Hunan First Normal University, Changsha, 410205, Hunan, P. R. China

    Hongxia Shi

  2. School of Mathematics and Statistics, Central South University, Changsha, 410083, Hunan, P. R. China

    Haibo Chen

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  1. Hongxia Shi
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  2. Haibo Chen
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Corresponding authors

Correspondence to Hongxia Shi or Haibo Chen.

Additional information

Hongxia Shi was supported by the National Natural Science Foundation of China (Grant No. 11801160) and the Natural Science Foundation of Hunan Province (Grant No. 2019JJ50096); Haibo Chen was supported by the National Natural Science Foundation of China (Grant No. 11671403).

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Shi, H., Chen, H. On a class of quasilinear Schrödinger equations with vanishing potentials and mixed nonlinearities. Indian J Pure Appl Math 50, 923–936 (2019). https://doi.org/10.1007/s13226-019-0364-1

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