JournalsrmiVol. 36, No. 5pp. 1549–1570

Ground state solutions for quasilinear Schrödinger equations with variable potential and superlinear reaction

  • Sitong Chen

    Central South University, Changsha, Hunan, China
  • Vicenţiu D. Rădulescu

    AGH University of Science and Technlogy, Kraków, Poland and Romanian Academy, Bucharest, Romania
  • Xianhua Tang

    Central South University, Changsha, Hunan, China
  • Binlin Zhang

    Shandong University of Science and Technology, Qingdao, and Heilongjiang Institute of Technology, Harbin, China
Ground state solutions for quasilinear Schrödinger equations with variable potential and superlinear reaction cover

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Abstract

This paper is concerned with the following quasilinear Schrödinger equation:

Δu+V(x)u12Δ(u2)u=g(u),xRN,-\Delta u+V(x)u-\frac{1}{2}\Delta (u^2)u= g(u), \quad x\in \mathbb{R}^N,

where N3N\ge 3, VC(RN,[0,))V\in \mathcal{C}(\mathbb R^N,[0,\infty)) and gC(R,R)g\in \mathcal{C}(\mathbb{R}, \mathbb{R}) is superlinear at infinity. By using variational and some new analytic techniques, we prove the above problem admits ground state solutions under mild assumptions on VV and gg. Moreover, we establish a minimax characterization of the ground state energy. Especially, we impose some new conditions on VV and more general assumptions on gg. For this, some new tricks are introduced to overcome the competing effect between the quasilinear term and the superlinear reaction. Hence our results improve and extend recent theorems in several directions.

Cite this article

Sitong Chen, Vicenţiu D. Rădulescu, Xianhua Tang, Binlin Zhang, Ground state solutions for quasilinear Schrödinger equations with variable potential and superlinear reaction. Rev. Mat. Iberoam. 36 (2020), no. 5, pp. 1549–1570

DOI 10.4171/RMI/1175