具有临界指数的分数阶Choquard方程非平凡非负解的多解性和集中性
Multiple and concentration of nontrivial nonnegative solutions for a fractional Choquard equation with critical exponent
摘要：
本文研究了分数阶Choquard方程$$\varepsilon^{2s}(-Δ)^s u+V(x)u=\varepsilon^{μ-N}(\frac{1}{|x|^μ}\ast F(u))f(u)+|u|^{2^\ast_s-2}u$$，其中$\varepsilon>0$是参数，$s\in(0,1),$，$N>2s,$，$2^*_s=\frac{2N}{N-2s}$和$0<μ<\min\{2s,N-2s\}$ .在对$V$和$f$适当的假设下，证明了该问题存在非平凡的非负基态解.此外，我们还将非平凡非负解的个数与位势达到最小值的集合的拓扑结构以及它们的集中行为联系起来。
英文摘要：
In present paper, we study the fractional Choquard equation $$\varepsilon^{2s}(-Δ)^s u+V(x)u=\varepsilon^{μ-N}(\frac{1}{|x|^μ}\ast F(u))f(u)+|u|^{2^\ast_s-2}u$$ where $\varepsilon>0$ is a parameter, $s\in(0,1),$ $N>2s,$ $2^*_s=\frac{2N}{N-2s}$ and $0<μ<\min\{2s,N-2s\}$. Under suitable assumption on $V$ and $f$, we prove this problem has a nontrivial nonnegative ground state solution. Moreover, we relate the number of nontrivial nonnegative solutions with the topology of the set where the potential attains its minimum values and their's concentration behavior.
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