Solutions to the coupled Schrödinger systems with steep potential well and critical exponent

Abstract In the present paper, we consider the coupled Schrödinger systems with critical exponent: <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="block" overflow="scroll"> <m:mfenced close="" open="{"> <m:mrow> <m:mtable class="cases"> <m:mtr> <m:mtd columnalign="left"> <m:mo>−</m:mo> <m:mi mathvariant="normal">Δ</m:mi> <m:msub> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:mi>i</m:mi> </m:mrow> </m:msub> <m:mo>+</m:mo> <m:mfenced close=")" open="("> <m:mrow> <m:mi>λ</m:mi> <m:msub> <m:mrow> <m:mi>V</m:mi> </m:mrow> <m:mrow> <m:mi>i</m:mi> </m:mrow> </m:msub> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>+</m:mo> <m:msub> <m:mrow> <m:mi>a</m:mi> </m:mrow> <m:mrow> <m:mi>i</m:mi> </m:mrow> </m:msub> </m:mrow> </m:mfenced> <m:msub> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:mi>i</m:mi> </m:mrow> </m:msub> <m:mo>=</m:mo> <m:munderover accent="true" accentunder="false"> <m:mrow> <m:mo>∑</m:mo> </m:mrow> <m:mrow> <m:mi>j</m:mi> <m:mo>=</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mrow> <m:mi>d</m:mi> </m:mrow> </m:munderover> <m:msub> <m:mrow> <m:mi>β</m:mi> </m:mrow> <m:mrow> <m:mi>i</m:mi> <m:mi>j</m:mi> </m:mrow> </m:msub> <m:msup> <m:mrow> <m:mfenced close="|" open="|"> <m:mrow> <m:msub> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:mi>j</m:mi> </m:mrow> </m:msub> </m:mrow> </m:mfenced> </m:mrow> <m:mrow> <m:mn>3</m:mn> </m:mrow> </m:msup> <m:mfenced close="|" open="|"> <m:mrow> <m:msub> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:mi>i</m:mi> </m:mrow> </m:msub> </m:mrow> </m:mfenced> <m:msub> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:mi>i</m:mi> </m:mrow> </m:msub> <m:mtext> </m:mtext> <m:mtext> in </m:mtext> <m:msup> <m:mrow> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> <m:mrow> <m:mn>3</m:mn> </m:mrow> </m:msup> <m:mo>,</m:mo> <m:mspace width="1em"/> </m:mtd> </m:mtr> <m:mtr> <m:mtd columnalign="left"> <m:msub> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:mi>i</m:mi> </m:mrow> </m:msub> <m:mo>∈</m:mo> <m:msup> <m:mrow> <m:mi>H</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msup> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:msup> <m:mrow> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> <m:mrow> <m:mi>N</m:mi> </m:mrow> </m:msup> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>,</m:mo> <m:mspace width="1em"/> <m:mi>i</m:mi> <m:mo>=</m:mo> <m:mn>1,2</m:mn> <m:mo>,</m:mo> <m:mo>…</m:mo> <m:mo>,</m:mo> <m:mi>d</m:mi> <m:mo>,</m:mo> <m:mspace width="1em"/> </m:mtd> </m:mtr> </m:mtable> </m:mrow> </m:mfenced> </m:math> $$\begin{cases}-{\Delta}{u}_{i}+\left(\lambda {V}_{i}\left(x\right)+{a}_{i}\right){u}_{i}=\sum _{j=1}^{d}{\beta }_{ij}{\left\vert {u}_{j}\right\vert }^{3}\left\vert {u}_{i}\right\vert {u}_{i}\quad \,\text{in}\,{\mathbb{R}}^{3},\quad \hfill \ {u}_{i}\in {H}^{1}\left({\mathbb{R}}^{N}\right)