Symmetry and monotonicity of positive solutions for a Choquard equation involving the logarithmic Laplacian operator. (English) Zbl 07925110
Summary: In this paper, we study a Schrödinger-Choquard equation involving the logarithmic Laplacian operator in \(\mathbb{R}^n\):
\[
\mathcal{L}_{\triangle} u(x)+\omega u(x)=C_{n,s}(|x|^{2s-n}*u^p) u^r, x\in \mathbb{R}^n,
\]
where \(0<s<1, p>1, r>0, n\geq 2, \omega >0\). Using the direct method of moving planes, we prove that if \(u\) satisfies some suitable asymptotic properties, then \(u\) must be radially symmetric and monotone decreasing about some point in the whole space. The key ingredients of the proofs are the narrow region principle and decay at infinity theorem; the ideas can be applied to problems involving more general nonlocal operators.
MSC:
| 35S05 | Pseudodifferential operators as generalizations of partial differential operators |
| 35B06 | Symmetries, invariants, etc. in context of PDEs |
| 35R09 | Integro-partial differential equations |
Keywords:
Choquard equation; logarithmic Laplacian operator; narrow region principle; decay at infinity; direct method of moving planesReferences:
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