
Ground States for Singularly Perturbed Planar Choquard Equation with Critical Exponential Growth 

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KeyWord:Choquard equation, critical exponential growth, TrudingerMoser inequality, ground state solution 
Author Name  Affiliation  Limin Zhang  Department of Mathematics, Suzhou University of Science and Technology, Suzhou, Jiangsu 215009, China HNPLAMA, 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  HNPLAMA, School of Mathematics and Statistics, Central South University, Changsha, Hunan 410083, China  Dongdong Qin  HNPLAMA, 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 TrudingerMoser 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. 


