Groundstate for the Schrödinger-Poisson-Slater equation involving the Coulomb-Sobolev critical exponent
Chun‐Yu Lei, Jun Lei, Hong-Min Suo
Abstract
Abstract In this article, we study the existence of ground state solutions for the Schrödinger-Poisson-Slater type equation with the Coulomb-Sobolev critical growth: <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:mfenced open="(" close=")"> <m:mrow> <m:mfrac> <m:mrow> <m:mn>1</m:mn> </m:mrow> <m:mrow> <m:mn>4</m:mn> <m:mi>π</m:mi> <m:mo>∣</m:mo> <m:mi>x</m:mi> <m:mo>∣</m:mo> </m:mrow> </m:mfrac> <m:mo>∗</m:mo> <m:mo>∣</m:mo> <m:mi>u</m:mi> <m:msup> <m:mrow> <m:mo>∣</m:mo> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msup> </m:mrow> </m:mfenced> <m:mi>u</m:mi> <m:mo>=</m:mo> <m:mo>∣</m:mo> <m:mi>u</m:mi> <m:mo>∣</m:mo> <m:mi>u</m:mi> <m:mo>+</m:mo> <m:mi>μ</m:mi> <m:mo>∣</m:mo> <m:mi>u</m:mi> <m:msup> <m:mrow> <m:mo>∣</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:mi>u</m:mi> <m:mo>,</m:mo> <m:mspace width="1.0em"/> <m:mi mathvariant="normal">in</m:mi> <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+\left(\frac{1}{4\pi | x| }\ast | u{| }^{2}\right)u=| u| u+\mu | u{| }^{p-2}u,\hspace{1.0em}{\rm{in}}\hspace{0.33em}{{\mathbb{R}}}^{3}, where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>μ</m:mi> <m:mo>></m:mo> <m:mn>0</m:mn> </m:math> \mu \gt 0 and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mn>3</m:mn> <m:mo><</m:mo> <m:mi>p</m:mi> <m:mo><</m:mo> <m:mn>6</m:mn> </m:math> 3\lt p\lt 6 . With the help of the Nehari-Pohozaev method, we obtain a ground-state solution for the above equation by employing compactness arguments.