Electron. J. Differential Equations, Vol. 2018 (2018), No. 63, pp. 1-21.

Existence of positive solutions to the nonlinear Choquard equation with competing potentials

Jun Wang, Mengmeng Qu, Lu Xiao

Abstract:
This article concerns the existence of positive solutions of the nonlinear Choquard equation
$$ -\Delta u+a(x)u=b(x)\Big(\frac{1}{|x|}*|u|^2\Big)u,\quad u\in H^{1}({\mathbb R}^3), $$
where the coefficients a and b are positive functions such that $a(x)\to\kappa_\infty$ and $b(x)\to \mu_\infty$ as $|x|\to\infty$. By comparing the decay rate of the coefficients a and b, we prove the existence of positive ground and bound stat solutions of Choquard equation.

Submitted October 11, 2017. Published March 7, 2018.
Math Subject Classifications: 35J61, 35J20, 35Q55, 49J40.
Key Words: Positive solutions; Choquard equation; competing coefficients; variational methods.

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Jun Wang
Faculty of Science
Jiangsu University
Zhenjiang, Jiangsu 212013, China
email: wangmath2011@126.com
Mengmeng Qu
Faculty of Science
Jiangsu University
Zhenjiang, Jiangsu 212013, China
email: qumengmengab@126.com
Lu Xiao
School of management
Jiangsu University
Zhenjiang, Jiangsu 212013, China
email: hnlulu@126.com

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