Abstract:

In this paper, we study the following fractional Choquard problem with shifting subcritical perturbation on bounded domains $$\begin{equation*} \left\{ \begin{aligned} &(-\Delta)^s u=\left(\int_{\Omega} \frac{u^{2^*_{\mu,s}}(y)}{|x-y|^\mu} \mathrm{d} y\right) u^{2^*_{\mu,s}-1}+g(x)\left[(u-k)^{+}\right]^{q-1}, &&x \in \Omega, \\ &u>0,\hspace{21em} &&x \in \Omega, \\ &u=0,\hspace{21em} &&x \in \mathbb{R}^N\backslash\Omega, \end{aligned} \right. \end{equation*}$$ where $N \geq 3$, $0<\mu<N$, $2^*_{\mu,s}=\frac{2N-\mu}{N-2s}$ is the fractional critical exponent in the sense of Hardy-Little-Wood-Sobolev inequality. Since the Choquard equation has non-local operator, we prove the existence of nontrivial solution $u_k$ for any $k\in(0,\infty)$ by energy estimation and variational method. What's more, the solutions $u_k$ are uniformly bounded when $k\to \infty$. At last, we get the concentration property of solutions.

Key words: fractional Choquard problem, subcritical perturbation, nontrivial solution, concentration