Concentration results for a magnetic Schrödinger-Poisson system with critical growth
Jingjing Liu, Chao Ji
Abstract
Abstract This paper is concerned with the following nonlinear magnetic Schrödinger-Poisson type equation <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:mfenced open="{" close=""> <m:mtable columnalign="right left right left right left right left right left right left" rowspacing=".5em" columnspacing="0.278em 2em 0.278em 2em 0.278em 2em 0.278em 2em 0.278em 2em 0.278em"> <m:mtr> <m:mtd/> <m:mtd> <m:mrow class="MJX-TeXAtom-ORD"> <m:mo maxsize="1.623em" minsize="1.623em">(</m:mo> </m:mrow> <m:mfrac> <m:mi>ϵ</m:mi> <m:mi>i</m:mi> </m:mfrac> <m:mi mathvariant="normal">∇</m:mi> <m:mo>−</m:mo> <m:mi>A</m:mi> <m:mo stretchy="false">(</m:mo> <m:mi>x</m:mi> <m:mo stretchy="false">)</m:mo> <m:msup> <m:mrow class="MJX-TeXAtom-ORD"> <m:mo maxsize="1.623em" minsize="1.623em">)</m:mo> </m:mrow> <m:mrow class="MJX-TeXAtom-ORD"> <m:mn>2</m:mn> </m:mrow> </m:msup> <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:msup> <m:mi>ϵ</m:mi> <m:mrow class="MJX-TeXAtom-ORD"> <m:mo>−</m:mo> <m:mn>2</m:mn> </m:mrow> </m:msup> <m:mo stretchy="false">(</m:mo> <m:mo fence="false" stretchy="false">|</m:mo> <m:mi>x</m:mi> <m:msup> <m:mo fence="false" stretchy="false">|</m:mo> <m:mrow class="MJX-TeXAtom-ORD"> <m:mo>−</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msup> <m:mo>∗</m:mo> <m:mo fence="false" stretchy="false">|</m:mo> <m:mi>u</m:mi> <m:msup> <m:mo fence="false" stretchy="false">|</m:mo> <m:mrow class="MJX-TeXAtom-ORD"> <m:mn>2</m:mn> </m:mrow> </m:msup> <m:mo stretchy="false">)</m:mo> <m:mi>u</m:mi> <m:mo>=</m:mo> <m:mi>f</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:mn>2</m:mn> </m:mrow> </m:msup> <m:mo stretchy="false">)</m:mo> <m:mi>u</m:mi> <m:mo>+</m:mo> <m:mo fence="false" stretchy="false">|</m:mo> <m:mi>u</m:mi> <m:msup> <m:mo fence="false" stretchy="false">|</m:mo> <m:mrow class="MJX-TeXAtom-ORD"> <m:mn>4</m:mn> </m:mrow> </m:msup> <m:mi>u</m:mi> <m:mspace width="1em"/> <m:mtext>in </m:mtext> <m:msup> <m:mrow class="MJX-TeXAtom-ORD"> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> <m:mn>3</m:mn> </m:msup> <m:mo>,</m:mo> </m:mtd> </m:mtr> <m:mtr> <m:mtd/> <m:mtd> <m:mi>u</m:mi> <m:mo>∈</m:mo> <m:msup> <m:mi>H</m:mi> <m:mrow class="MJX-TeXAtom-ORD"> <m:mn>1</m:mn> </m:mrow> </m:msup> <m:mo stretchy="false">(</m:mo> <m:msup> <m:mrow class="MJX-TeXAtom-ORD"> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> <m:mrow class="MJX-TeXAtom-ORD"> <m:mn>3</m:mn> </m:mrow> </m:msup> <m:mo>,</m:mo> <m:mrow class="MJX-TeXAtom-ORD"> <m:mi mathvariant="double-struck">C</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> <m:mo>,</m:mo> </m:mtd> </m:mtr> </m:mtable> </m:mfenced> </m:mstyle> </m:mtd> </m:mtr> </m:mtable> </m:math> $$\begin{array}{} \displaystyle \left\{ \begin{aligned} &\Big(\frac{\epsilon}{i}\nabla-A(x)\Big)^{2}u+V(x)u+\epsilon^{-2}(\vert x\vert^{-1}\ast \vert u\vert^{2})u=f(|u|^{2})u+\vert u\vert^{4}u \quad \hbox{in }\mathbb{R}^3,\\ &u\in H^{1}(\mathbb{R}^{3}, \mathbb{C}), \end{aligned} \right. \end{array}$$ where ϵ > 0, V : ℝ 3 → ℝ and A</jats