Multiplicity and concentration behaviour of solutions for a fractional Choquard equation with critical growth
Zhipeng Yang, Fukun Zhao
Abstract
Abstract In this paper, we study the singularly perturbed fractional Choquard equation <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="block"> <m:msup> <m:mi>ε</m:mi> <m:mrow class="MJX-TeXAtom-ORD"> <m:mn>2</m:mn> <m:mi>s</m:mi> </m:mrow> </m:msup> <m:mo stretchy="false">(</m:mo> <m:mo>−</m:mo> <m:mrow class="MJX-TeXAtom-ORD"> <m:mi class="MJX-tex-mathit" mathvariant="italic">Δ</m:mi> </m:mrow> <m:msup> <m:mo stretchy="false">)</m:mo> <m:mi>s</m:mi> </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:mi>μ</m:mi> <m:mo>−</m:mo> <m:mn>3</m:mn> </m:mrow> </m:msup> <m:mo stretchy="false">(</m:mo> <m:munder> <m:mo>∫</m:mo> <m:mrow> <m:msup> <m:mrow> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> <m:mn>3</m:mn> </m:msup> </m:mrow> </m:munder> <m:mfrac> <m:mrow> <m:mrow class="MJX-TeXAtom-ORD"> <m:mo stretchy="false">|</m:mo> </m:mrow> <m:mi>u</m:mi> <m:mo stretchy="false">(</m:mo> <m:mi>y</m:mi> <m:mo stretchy="false">)</m:mo> <m:msup> <m:mrow class="MJX-TeXAtom-ORD"> <m:mo stretchy="false">|</m:mo> </m:mrow> <m:mrow class="MJX-TeXAtom-ORD"> <m:msubsup> <m:mn>2</m:mn> <m:mrow class="MJX-TeXAtom-ORD"> <m:mi>μ</m:mi> <m:mo>,</m:mo> <m:mi>s</m:mi> </m:mrow> <m:mo>∗</m:mo> </m:msubsup> </m:mrow> </m:msup> <m:mo>+</m:mo> <m:mi>F</m:mi> <m:mo stretchy="false">(</m:mo> <m:mi>u</m:mi> <m:mo stretchy="false">(</m:mo> <m:mi>y</m:mi> <m:mo stretchy="false">)</m:mo> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mrow> <m:mrow class="MJX-TeXAtom-ORD"> <m:mo stretchy="false">|</m:mo> </m:mrow> <m:mi>x</m:mi> <m:mo>−</m:mo> <m:mi>y</m:mi> <m:msup> <m:mrow class="MJX-TeXAtom-ORD"> <m:mo stretchy="false">|</m:mo> </m:mrow> <m:mi>μ</m:mi> </m:msup> </m:mrow> </m:mfrac> <m:mi>d</m:mi> <m:mi>y</m:mi> <m:mo stretchy="false">)</m:mo> <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:msubsup> <m:mn>2</m:mn> <m:mrow class="MJX-TeXAtom-ORD"> <m:mi>μ</m:mi> <m:mo>,</m:mo> <m:mi>s</m:mi> </m:mrow> <m:mo>∗</m:mo> </m:msubsup> <m:mo>−</m:mo> <m:mn>2</m:mn> </m:mrow> </m:msup> <m:mi>u</m:mi> <m:mo>+</m:mo> <m:mfrac> <m:mn>1</m:mn> <m:msubsup> <m:mn>2</m:mn> <m:mrow class="MJX-TeXAtom-ORD"> <m:mi>μ</m:mi> <m:mo>,</m:mo> <m:mi>s</m:mi> </m:mrow> <m:mo>∗</m:mo> </m:msubsup> </m:mfrac> <m:mi>f</m:mi> <m:mo stretchy="false">(</m:mo> <m:mi>u</m:mi> <m:mo stretchy="false">)</m:mo> <m:mo stretchy="false">)</m:mo> <m:mspace width="thinmathspace"/> <m:mtext>in</m:mtext> <m:mspace width="thinmathspace"/> <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:math> $$\begin{equation*}\varepsilon^{2s}(-{\it\Delta})^su+V(x)u=\varepsilon^{\mu-3}(\int\limits_{\mathbb{R}^3}\frac{|u(y)|^{2^*_{\mu,s}}+F(u(y))}{|x-y|^\mu}dy)(|u|^{2^*_{\mu,s}-2}u+\frac{1}{2^*_{\mu,s}}f(u)) \, \text{in}\, \mathbb{R}^3, \end{equation*}$$ where ε > 0 is a small parameter, (− △ ) s denotes the fractional Laplacian of order s ∈ (0, 1), 0 < μ < 3, <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msubsup> <m:mn>2</m:mn> <m:mrow> <m:mi>μ</m:mi> <m:mo>,</m:mo> <m:mi>s</m:mi> </m:mrow> <m:mo>⋆</m:mo> </m:msubsup> <m:mo>=</m:mo> <m:mfrac> <m:mrow> <m:mn>6</m:mn> <m:mo>−</m:mo> <m:mi>μ</m:mi> </m:mrow> <m:mrow> <m:mn>3</m:mn> <m:mo>−</m:mo> <m:mn>2</m:mn> <m:mi>s</m:mi> </m:mrow> </m:mfrac> </m:mrow>