Groundstates for Choquard type equations with weighted potentials and Hardy–Littlewood–Sobolev lower critical exponent
Shuai Zhou, Zhisu Liu, Jianjun Zhang
Abstract
Abstract We are concerned with a class of Choquard type equations with weighted potentials and Hardy–Littlewood–Sobolev lower critical exponent <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="block"> <m:mtable rowspacing="4pt" columnspacing="1em"> <m:mtr> <m:mtd> <m:mstyle displaystyle="true"> <m:mo>−</m:mo> <m:mrow class="MJX-TeXAtom-ORD"> <m:mi class="MJX-tex-mathit" mathvariant="italic">Δ</m:mi> </m:mrow> <m:mi>u</m:mi> <m:mo>+</m:mo> <m:mi>V</m:mi> <m:mo stretchy="false">(</m:mo> <m:mi>x</m:mi> <m:mo stretchy="false">)</m:mo> <m:mi>u</m:mi> <m:mo>=</m:mo> <m:mfenced open="(" close=")"> <m:mrow> <m:msub> <m:mi>I</m:mi> <m:mrow class="MJX-TeXAtom-ORD"> <m:mi>α</m:mi> </m:mrow> </m:msub> <m:mo>∗</m:mo> <m:mo stretchy="false">[</m:mo> <m:mi>Q</m:mi> <m:mo stretchy="false">(</m:mo> <m:mi>x</m:mi> <m:mo stretchy="false">)</m:mo> <m:mrow class="MJX-TeXAtom-ORD"> <m:mo stretchy="false">|</m:mo> </m:mrow> <m:mi>u</m:mi> <m:msup> <m:mrow class="MJX-TeXAtom-ORD"> <m:mo stretchy="false">|</m:mo> </m:mrow> <m:mrow class="MJX-TeXAtom-ORD"> <m:mfrac> <m:mrow> <m:mi>N</m:mi> <m:mo>+</m:mo> <m:mi>α</m:mi> </m:mrow> <m:mi>N</m:mi> </m:mfrac> </m:mrow> </m:msup> <m:mo stretchy="false">]</m:mo> </m:mrow> </m:mfenced> <m:mi>Q</m:mi> <m:mo stretchy="false">(</m:mo> <m:mi>x</m:mi> <m:mo stretchy="false">)</m:mo> <m:mrow class="MJX-TeXAtom-ORD"> <m:mo stretchy="false">|</m:mo> </m:mrow> <m:mi>u</m:mi> <m:msup> <m:mrow class="MJX-TeXAtom-ORD"> <m:mo stretchy="false">|</m:mo> </m:mrow> <m:mrow class="MJX-TeXAtom-ORD"> <m:mfrac> <m:mi>α</m:mi> <m:mi>N</m:mi> </m:mfrac> <m:mo>−</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msup> <m:mi>u</m:mi> <m:mo>,</m:mo> <m:mspace width="1em"/> <m:mi>x</m:mi> <m:mo>∈</m:mo> <m:msup> <m:mrow class="MJX-TeXAtom-ORD"> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> <m:mi>N</m:mi> </m:msup> <m:mo>.</m:mo> </m:mstyle> </m:mtd> </m:mtr> </m:mtable> </m:math> $$\begin{array}{} \displaystyle -{\it\Delta} u+V(x)u=\left(I_{\alpha}\ast [Q(x)|u|^{\frac{N+\alpha}{N}}]\right)Q(x)|u|^{\frac{\alpha}{N}-1}u, \quad x\in \mathbb R^N. \end{array}$$ By using variational approaches, we investigate the existence of groundstates relying on the asymptotic behaviour of weighted potentials at infinity. Moreover, non-existence of non-trivial solutions is also considered. In particular, we give a partial answer to some open questions raised in [D.~Cassani, J. Van Schaftingen and J. J. Zhang, Groundstates for Choquard type equations with Hardy-Littlewood-Sobolev lower critical exponent, Proceedings of the Royal Society of Edinburgh, Section A Mathematics , 150 (2020), 1377–1400].