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

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).

Keywords

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.

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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