Limit profiles and uniqueness of ground states to the nonlinear Choquard equations

Abstract

Abstract Consider nonlinear Choquard equations <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>{</m:mo> <m:mtable columnspacing="0pt" displaystyle="true" rowspacing="0pt"> <m:mtr> <m:mtd columnalign="right"> <m:mrow> <m:mrow> <m:mo>-</m:mo> <m:mrow> <m:mi mathvariant="normal">Δ</m:mi> <m:mo>⁢</m:mo> <m:mi>u</m:mi> </m:mrow> </m:mrow> <m:mo>+</m:mo> <m:mi>u</m:mi> </m:mrow> </m:mtd> <m:mtd columnalign="left"> <m:mrow> <m:mi/> <m:mo>=</m:mo> <m:mrow> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:msub> <m:mi>I</m:mi> <m:mi>α</m:mi> </m:msub> <m:mo>*</m:mo> <m:msup> <m:mrow> <m:mo fence="true" stretchy="false">|</m:mo> <m:mi>u</m:mi> <m:mo fence="true" stretchy="false">|</m:mo> </m:mrow> <m:mi>p</m:mi> </m:msup> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>⁢</m:mo> <m:msup> <m:mrow> <m:mo fence="true" stretchy="false">|</m:mo> <m:mi>u</m:mi> <m:mo fence="true" stretchy="false">|</m:mo> </m:mrow> <m:mrow> <m:mi>p</m:mi> <m:mo>-</m:mo> <m:mn>2</m:mn> </m:mrow> </m:msup> <m:mo>⁢</m:mo> <m:mi>u</m:mi> </m:mrow> </m:mrow> </m:mtd> <m:mtd/> <m:mtd columnalign="right"> <m:mrow> <m:mrow> <m:mtext>in </m:mtext> <m:mo>⁢</m:mo> <m:msup> <m:mi>ℝ</m:mi> <m:mi>N</m:mi> </m:msup> </m:mrow> <m:mo>,</m:mo> </m:mrow> </m:mtd> </m:mtr> <m:mtr> <m:mtd columnalign="right"> <m:mrow> <m:munder> <m:mo movablelimits="false">lim</m:mo> <m:mrow> <m:mi>x</m:mi> <m:mo>→</m:mo> <m:mi mathvariant="normal">∞</m:mi> </m:mrow> </m:munder> <m:mo>⁡</m:mo> <m:mrow> <m:mi>u</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>x</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:mtd> <m:mtd columnalign="left"> <m:mrow> <m:mrow> <m:mi/> <m:mo>=</m:mo> <m:mn>0</m:mn> </m:mrow> <m:mo>,</m:mo> </m:mrow> </m:mtd> </m:mtr> </m:mtable> </m:mrow> </m:math> \left\{\begin{aligned} \displaystyle-\Delta u+u&amp;\displaystyle=(I_{\alpha}*% \lvert u\rvert^{p})\lvert u\rvert^{p-2}u&amp;&amp;\displaystyle\phantom{}\text{in }% \mathbb{R}^{N},\\ \displaystyle\lim_{x\to\infty}u(x)&amp;\displaystyle=0,\end{aligned}\right. where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>I</m:mi> <m:mi>α</m:mi> </m:msub> </m:math> {I_{\alpha}} denotes the Riesz potential and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>α</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mn>0</m:mn> <m:mo>,</m:mo> <m:mi>N</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {\alpha\in(0,N)} . In this paper, we investigate limit profiles of ground states of nonlinear Choquard equations as <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>α</m:mi> <m:mo>→</m:mo> <m:mn>0</m:mn> </m:mrow> </m:math> {\alpha\to 0} or <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>α</m:mi> <m:mo>→</m:mo> <m:mi>N</m:mi> </m:mrow> </m:math>