Infinitely many non-radial solutions for a Choquard equation
Fashun Gao, Minbo Yang
Abstract
Abstract In this article, we consider the non-linear Choquard equation <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="block"> <m:mo>−</m:mo> <m:mi mathvariant="normal">Δ</m:mi> <m:mi>u</m:mi> <m:mo>+</m:mo> <m:mi>V</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mo>∣</m:mo> <m:mi>x</m:mi> <m:mo>∣</m:mo> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mi>u</m:mi> <m:mo>=</m:mo> <m:mfenced open="(" close=")"> <m:mrow> <m:munder> <m:mrow> <m:mrow> <m:mo>∫</m:mo> </m:mrow> </m:mrow> <m:mrow> <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:mrow> </m:munder> <m:mfrac> <m:mrow> <m:mo>∣</m:mo> <m:mi>u</m:mi> <m:mrow> <m:mrow> <m:mo>(</m:mo> </m:mrow> <m:mrow> <m:mi>y</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:msup> <m:mrow> <m:mo>∣</m:mo> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msup> </m:mrow> <m:mrow> <m:mo>∣</m:mo> <m:mi>x</m:mi> <m:mo>−</m:mo> <m:mi>y</m:mi> <m:mo>∣</m:mo> </m:mrow> </m:mfrac> <m:mi mathvariant="normal">d</m:mi> <m:mi>y</m:mi> </m:mrow> </m:mfenced> <m:mi>u</m:mi> <m:mspace width="1.0em"/> <m:mstyle> <m:mspace width="0.1em"/> <m:mtext>in</m:mtext> <m:mspace width="0.1em"/> </m:mstyle> <m:mspace width="0.33em"/> <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:math> -\Delta u+V\left(| x| )u=\left(\mathop{\int }\limits_{{{\mathbb{R}}}^{3}}\frac{| u(y){| }^{2}}{| x-y| }{\rm{d}}y\right)u\hspace{1.0em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{3}, where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>V</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>r</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> V\left(r) is a positive bounded function. Under some proper assumptions on <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>V</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>r</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> V\left(r) , we are able to establish the existence of infinitely many non-radial solutions.