Nehari-type ground state solutions for a Choquard equation with doubly critical exponents
Sitong Chen, Xianhua Tang, Jiuyang Wei
Abstract
Abstract This paper deals with the following Choquard equation with a local nonlinear perturbation: <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="left left" rowspacing="4pt" columnspacing="1em"> <m:mtr> <m:mtd> <m:mo>−</m:mo> <m:mrow class="MJX-TeXAtom-ORD"> <m:mi class="MJX-tex-mathit" mathvariant="italic">Δ</m:mi> </m:mrow> <m:mi>u</m:mi> <m:mo>+</m:mo> <m:mi>u</m:mi> <m:mo>=</m:mo> <m:mfenced open="(" close=")"> <m:mrow> <m:msub> <m:mi>I</m:mi> <m:mrow class="MJX-TeXAtom-ORD"> <m:mi>α</m:mi> </m:mrow> </m:msub> <m:mo>∗</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:mfrac> <m:mi>α</m:mi> <m:mn>2</m:mn> </m:mfrac> <m:mo>+</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msup> </m:mrow> </m:mfenced> <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:mfrac> <m:mi>α</m:mi> <m:mn>2</m:mn> </m:mfrac> <m:mo>−</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msup> <m:mi>u</m:mi> <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:mo>,</m:mo> </m:mtd> <m:mtd> <m:mi>x</m:mi> <m:mo>∈</m:mo> <m:msup> <m:mrow class="MJX-TeXAtom-ORD"> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> <m:mn>2</m:mn> </m:msup> <m:mo>;</m:mo> </m:mtd> </m:mtr> <m:mtr> <m:mtd> <m:mi>u</m:mi> <m:mo>∈</m:mo> <m:msup> <m:mi>H</m:mi> <m:mn>1</m:mn> </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:mn>2</m:mn> </m:msup> <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> $$\begin{array}{} \displaystyle \left\{ \begin{array}{ll} - {\it\Delta} u+u=\left(I_{\alpha}*|u|^{\frac{\alpha}{2}+1}\right)|u|^{\frac{\alpha}{2}-1}u +f(u), & x\in \mathbb{R}^2; \\ u\in H^1(\mathbb{R}^2), \end{array} \right. \end{array}$$ where α ∈ (0, 2), I α : ℝ 2 → ℝ is the Riesz potential and f ∈ 𝓒(ℝ, ℝ) is of critical exponential growth in the sense of Trudinger-Moser. The exponent <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mtable rowspacing="4pt" columnspacing="1em"> <m:mtr> <m:mtd> <m:mstyle displaystyle="true"> <m:mfrac> <m:mi>α</m:mi> <m:mn>2</m:mn> </m:mfrac> <m:mo>+</m:mo> <m:mn>1</m:mn> </m:mstyle> </m:mtd> </m:mtr> </m:mtable> </m:math> $\begin{array}{} \displaystyle \frac{\alpha}{2}+1 \end{array}$ is critical with respect to the Hardy-Littlewood-Sobolev inequality. We obtain the existence of a nontrivial solution or a Nehari-type ground state solution for the above equation in the doubly critical case, i.e. the appearance of both the lower critical exponent <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mtable rowspacing="4pt" columnspacing="1em"> <m:mtr> <m:mtd> <m:mstyle displaystyle="true"> <m:mfrac>