Electron. J. Differential Equations, Vol. 2018 (2018), No. 172, pp. 1-27.

Multiple solutions for nonhomogeneous Choquard equations

Lixia Wang

Abstract:
In this article, we consider the multiple solutions for the nonhomogeneous Choquard equations
$$ - \Delta u +u=\Big(\frac{1}{|x|^{\alpha}}\ast |u|^{p}\Big)|u|^{p-2}u+h(x), \quad x\in \mathbb{R}^N, $$
and
$$ - \Delta u=\Big(\frac{1}{|x|^{\alpha}}\ast |u|^{2^{\ast}_{\alpha}} \Big)|u|^{2^{\ast}_{\alpha}-2}u+h(x), \quad x\in \mathbb{R}^N, $$
where $N\geq 3$, $0<\alpha<N$, $2-\frac{\alpha}{N}<p<2^{\ast}_{\alpha}=\frac{2N-\alpha}{N-2}$. Under suitable assumptions on h, we obtain at least two solutions on the subcritical case $2-\frac{\alpha}{N}<p<2^{\ast}_{\alpha}$ and on the critical case $p=2^{\ast}_{\alpha}$.

Submitted August 2, 2017. Published October 17, 2018.
Math Subject Classifications: 35J20, 35J60.
Key Words: Choquard equation; nonhomogeneous; critical exponent.

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Lixia Wang
School of Sciences
Tianjin Chengjian University
Tianjin 300384, China
email: wanglixia0311@126.com

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