Multiple solutions for nonhomogeneous Choquard equations

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 $p=2^{\ast}_{\alpha}$.

Keywords

• Choquard equation
• nonhomogeneous
• critical exponent