Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
173 tokens/sec
GPT-4o
7 tokens/sec
Gemini 2.5 Pro Pro
46 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

High energy positive solutions for a coupled Hartree system with Hardy-Littlewood-Sobolev critical exponents (2009.03102v1)

Published 7 Sep 2020 in math.AP

Abstract: We study the coupled Hartree system $$ \left{\begin{array}{ll} -\Delta u+ V_1(x)u =\alpha_1\big(|x|{-4}\ast u{2}\big)u+\beta \big(|x|{-4}\ast v{2}\big)u &\mbox{in}\ \mathbb{R}N,\[1mm] -\Delta v+ V_2(x)v =\alpha_2\big(|x|{-4}\ast v{2}\big)v +\beta\big(|x|{-4}\ast u{2}\big)v &\mbox{in}\ \mathbb{R}N, \end{array}\right. $$ where $N\geq 5$, $\beta>\max{\alpha_1,\alpha_2}\geq\min{\alpha_1,\alpha_2}>0$, and $V_1,\,V_2\in L{N/2}(\mathbb{R}N)\cap L_{\text{loc}}{\infty}(\mathbb{R}N)$ are nonnegative potentials. This system is critical in the sense of the Hardy-Littlewood-Sobolev inequality. For the system with $V_1=V_2=0$ we employ moving sphere arguments in integral form to classify positive solutions and to prove the uniqueness of positive solutions up to translation and dilation, which is of independent interest. Then using the uniqueness property, we establish a nonlocal version of the global compactness lemma and prove the existence of a high energy positive solution for the system assuming that $|V_1|{L{N/2}(\mathbb{R}N)}+|V_2|{L{N/2}(\mathbb{R}N)}>0$ is suitably small.

Summary

We haven't generated a summary for this paper yet.