| Volume 15, Number 1, 2025, Pages - DOI：10.11948/JAAC-2023-0439 |
| Multiplicity and concentration of solutions to a singular Choquard equation with critical Sobolev exponent |
| Shengbin Yu,Jianqing Chen |
| Keywords:Singular Choquard equation Variational method Concentration Critical Sobolev exponent |
| Abstract: |
| In this paper, we consider a nonautonomous singular
Choquard equation with critical exponent
$$\left \{\begin{array}{lcl}
-\Delta u+V(x)u+\lambda(I_\alpha\ast |u|^{p})|u|^{p-2}u=f(x)u^{-\gamma}+|u|^{4}u, &&\quad
x\in\mathbb{R}^3,\ u>0,&&\quad x\in\mathbb{R}^3,
\end{array}\right.$$
where $I_\alpha$ is the Riesz potential of order $\alpha\in(0,3)$ and $1+\frac{\alpha}{3}\le p<3$, $0<\gamma<1$.
Under certain assumptions on $V$ and $f$, we show
the existence and multiplicity of
positive solutions for $\lambda>0$ by using variational method and Nehari type constraint. We also study concentration of solutions as $\lambda\rightarrow0^+$. |
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