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Ground States for Singularly Perturbed Planar Choquard Equation with Critical Exponential Growth
  
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KeyWord:Choquard equation, critical exponential growth, Trudinger-Moser inequality, ground state solution
Author NameAffiliation
Limin Zhang Department of Mathematics, Suzhou University of Science and Technology, Suzhou, Jiangsu 215009, China
HNP-LAMA, School of Mathematics and Statistics, Central South University, Changsha, Hunan 410083, China 
Fangfang Liao Department of Mathematics, Xiangnan University, Chenzhou, Hunan 423000, China 
Xianhua Tang HNP-LAMA, School of Mathematics and Statistics, Central South University, Changsha, Hunan 410083, China 
Dongdong Qin HNP-LAMA, School of Mathematics and Statistics, Central South University, Changsha, Hunan 410083, China 
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Abstract:
      In this paper, we are dedicated to studying the following singularly Choquard equation $$ -\varepsilon^2\Delta u+V(x)u=\varepsilon^{-\alpha}\left[I_{\alpha}\ast F(u)\right]f(u),\ \ \ \ x\in\R^2,$$ where $V(x)$ is a continuous real function on $\R^2$, $I_{\alpha}:\R^2\rightarrow\R$ is the Riesz potential, and $F$ is the primitive function of nonlinearity $f$ which has critical exponential growth. Using the Trudinger-Moser inequality and some delicate estimates, we show that the above problem admits at least one semiclassical ground state solution, for $\varepsilon>0$ small provided that $V(x)$ is periodic in $x$ or asymptotically linear as $|x|\rightarrow \infty$. In particular, a precise and fine lower bound of $\frac{f(t)}{e^{\beta_{0} t^{2}}}$ near infinity is introduced in this paper.