Positive solutions for a class of supercritical quasilinear Schrödinger equations

Yin Deng; Xiaojing Zhang; Gao Jia; Yin Deng; Xiaojing Zhang; Gao Jia

doi:10.3934/math.2022366

AIMS Mathematics

2022, Volume 7, Issue 4: 6565-6582. doi: 10.3934/math.2022366

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Research article

Positive solutions for a class of supercritical quasilinear Schrödinger equations

1.
Business School, University of Shanghai for Science and Technology, Shanghai, 200093, China
2.
School of Mathematical Sciences, Beijing Normal University, Beijing, 100875, China
3.
College of Science, University of Shanghai for Science and Technology, Shanghai, 200093, China

Received: 21 October 2021 Revised: 04 January 2022 Accepted: 07 January 2022 Published: 21 January 2022
MSC : 35B45, 35J20, 35J62

This paper deals with a class of supercritical quasilinear Schrödinger equations

$ -\Delta u+V(x)u+\kappa\Delta(\sqrt{1+{u}^{2}})\frac{u}{2\sqrt{1+{u}^{2}}} = \lambda f(u), \; x\in \mathbb{R}^{N}, $

where $ \kappa\geq2, \; N\geq3, \; \lambda > 0. $ We suppose that the nonlinearity $ f(t):\mathbb{R}\rightarrow \mathbb{R} $ is continuous and only superlinear in a neighbourhood of $ t = 0. $ By using a change of variable and the variational methods, we obtain the existence of positive solutions for the above problem.
- quasilinear Schrödinger equations,
- periodic potential,
- asymptotically periodic potential,
- variational methods,
- $ L^{\infty} $-estimates
Citation: Yin Deng, Xiaojing Zhang, Gao Jia. Positive solutions for a class of supercritical quasilinear Schrödinger equations[J]. AIMS Mathematics, 2022, 7(4): 6565-6582. doi: 10.3934/math.2022366

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Abstract

This paper deals with a class of supercritical quasilinear Schrödinger equations

$ -\Delta u+V(x)u+\kappa\Delta(\sqrt{1+{u}^{2}})\frac{u}{2\sqrt{1+{u}^{2}}} = \lambda f(u), \; x\in \mathbb{R}^{N}, $

where $ \kappa\geq2, \; N\geq3, \; \lambda > 0. $ We suppose that the nonlinearity $ f(t):\mathbb{R}\rightarrow \mathbb{R} $ is continuous and only superlinear in a neighbourhood of $ t = 0. $ By using a change of variable and the variational methods, we obtain the existence of positive solutions for the above problem.

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