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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 Name | Affiliation | | 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. |
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