Semi-classical states for a class of quasilinear Schrödinger equations with a parameter

Hongjie Xu; Xianyong Yang; Hongjie Xu; Xianyong Yang

doi:10.3934/era.2025314

Electronic Research Archive

2025, Volume 33, Issue 11: 7103-7125. doi: 10.3934/era.2025314

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Semi-classical states for a class of quasilinear Schrödinger equations with a parameter

Hongjie Xu ,
Xianyong Yang ^,

School of Mathematics and Computer Science, Yunnan Minzu University, Kunming 650504, China

Received: 07 October 2025 Revised: 11 November 2025 Accepted: 13 November 2025 Published: 24 November 2025

In this paper, we study the following quasilinear Schrödinger equation with a parameter: $ -\varepsilon^2\Delta u+V(x)u-\varepsilon^2\kappa\beta\Delta(|u|^{2\beta})|u|^{2\beta-2}u = |u|^{q-2}u+|u|^{(2\beta)2^*-2}u $ in $ \mathbb{R}^N $, where $ N\geqslant3 $, $ \beta > \frac{1}{2} $, $ 4\beta < q < (2\beta)2^* $, $ \kappa $ is a constant, and $ V:\mathbb{R}^N\to\mathbb{R} $ satisfies the classical global assumption. By using a change of variable, we obtain the existence, multiplicity, and concentration behavior. We will see that the existence of solutions depends heavily on the parameter $ \beta $.
- semi-classical states,
- quasilinear Schrödinger equations,
- a parameter,
- critical growth
Citation: Hongjie Xu, Xianyong Yang. Semi-classical states for a class of quasilinear Schrödinger equations with a parameter[J]. Electronic Research Archive, 2025, 33(11): 7103-7125. doi: 10.3934/era.2025314

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Abstract

In this paper, we study the following quasilinear Schrödinger equation with a parameter: $ -\varepsilon^2\Delta u+V(x)u-\varepsilon^2\kappa\beta\Delta(|u|^{2\beta})|u|^{2\beta-2}u = |u|^{q-2}u+|u|^{(2\beta)2^*-2}u $ in $ \mathbb{R}^N $, where $ N\geqslant3 $, $ \beta > \frac{1}{2} $, $ 4\beta < q < (2\beta)2^* $, $ \kappa $ is a constant, and $ V:\mathbb{R}^N\to\mathbb{R} $ satisfies the classical global assumption. By using a change of variable, we obtain the existence, multiplicity, and concentration behavior. We will see that the existence of solutions depends heavily on the parameter $ \beta $.

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