Dynamics of Partial Differential Equations

Volume 21 (2024)

Number 1

Liouville theorems for nonnegative solutions to weighted Schrödinger equations with logarithmic nonlinearities

Pages: 31 – 60

DOI: https://dx.doi.org/10.4310/DPDE.2024.v21.n1.a2


Yuxia Guo (Department of Mathematics, Tsinghua University, Beijing, China)

Shaolong Peng (Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, China)


In this paper, we are concerned with the physically interesting static weighted Schrödinger equations involving logarithmic nonlinearities:\[(-\Delta)^s u = c_1 {\lvert x \rvert}^a u^{p_1} \log (1 + u^{q_1}) + c_2 {\lvert x \rvert}^b\Bigl( \dfrac{1}{{\lvert \: \cdot \: \rvert}^\sigma} \Bigr) u^{p_2} \quad \textrm{,}\]where $n \geq 2, 0 \lt s =: m + \frac{\alpha}{2} \lt +\infty , 0 \lt \alpha \leq 2, 0 \lt \sigma \lt n, c_1, c_2 \geq 0$ with $c_1 + c_2 \gt 0, 0 \leq a, b \lt+\infty$.

Here we point out the above equations involving higher-order or higher-order fractional Laplacians. We first derive the Liouville theorems (i.e., non-existence of nontrivial nonnegative solutions) in the subcritical-order cases (see Theorem 1.1) via the method of scaling spheres. Secondly, we obtain the Liouville-type results in critical and supercritical-order cases (see Theorem 1.2) by using some integral inequalities. As applications, we also derive Liouville-type results for the Schrödinger system involving logarithmic nonlinearities (see Theorem 1.4).


higher-order fractional Laplacians, Schrödinger equations, logarithmic nonlinearities, nonnegative solutions, Liouville theorems, method of scaling spheres

2010 Mathematics Subject Classification

Primary 35B53. Secondary 35J30, 35R10.

Yuxia Guo was supported by NSFC (No. 12031015, 12271283). Shaolong Peng is supported by the NSFC (No. 11971049).

Received 14 March 2022

Published 7 November 2023