具一般对数非线性项分数阶薛定谔方程的基态解——献给李工宝教授 70 寿辰

Abstract

Abstract:

In this paper, we consider the following Schrödinger equations with general logarithmic nonlinear terms

$\begin{equation*} (-\Delta)^s u = u(\log|u|)^{\alpha}\ \text{in}\ \mathbb{R}^N, \end{equation*}$

where $0<s<1$, $N>2s$, $\alpha\ge 1$ is a constant. By observing the convergent phenomenon of the power-law Schrödinger equation $(-\Delta)^s u = u(|u|^{\sigma}-1)^{\alpha}$ as $\sigma\to 0^+$, we show that the problem has a positive ground state solution if $(-1)^{\alpha}=-1$.

Key words: fractional Schr?dinger equations, logarithmic nonlinear term, power-law, convergence, ground state solution.

CLC Number:

O175.2

Xiaoming An, Yining Fang, Zhengchang Jin. Ground State Solution for Fractional Schrödinger Equations with General Logarithmic Nonlinear Terms[J].Acta mathematica scientia,Series A, 2025, 45(6): 1839-1853.

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References 35

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Ground State Solution for Fractional Schrödinger Equations with General Logarithmic Nonlinear Terms

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