含对数型非线性项的拟线性薛定谔方程的驻波解——献给邓引斌教授 70 寿辰

Abstract

Abstract:

This study investigates the existence of non-trivial classical solutions for a class of parameterized quasilinear Schrödinger equations containing logarithmic nonlinear terms:

$ -\Delta u + V(x)u + \frac{\kappa}{2} [\Delta |u|^2 ]u = u\log (1 + |u|^2), \quad x\in \mathbb{R}^N $

where $N \geq 3$, $\kappa > 0$ is a parameter, and $V:\mathbb{R}^{N} \rightarrow \mathbb{R}$ is a continuous function. The model holds significant importance in plasma physics and nonlinear optics. By combining variational methods and perturbation techniques, we demonstrate that non-trivial solutions exist for sufficiently small parameters $\kappa$, and establish $L^{\infty}$ estimates for these solutions.

Key words: quasilinear Schr?dinger equation, logarithmic nonlinear term, mountain pass theorem, standing wave, variational method

CLC Number:

O175.23

Qingfei Jin. Standing Wave Solutions of the Quasilinear Schrödinger Equation with Logarithmic Nonlinear Terms[J].Acta mathematica scientia,Series A, 2026, 46(4): 1572-1584.

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References 16

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Standing Wave Solutions of the Quasilinear Schrödinger Equation with Logarithmic Nonlinear Terms

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