Issue 
ESAIM: COCV
Volume 24, Number 1, JanuaryMarch 2018



Page(s)  177  209  
DOI  https://doi.org/10.1051/cocv/2017007  
Published online  17 January 2018 
Semiclassical ground state solutions for a Choquard type equation in ℝ^{2} with critical exponential growth^{∗}
Department of Mathematics, Zhejiang Normal University, Jinhua, Zhejiang, 321004, P.R. China.
mbyang@zjnu.edu.cn
Received: 28 September 2016
Accepted: 25 January 2017
In this paper we study a nonlocal singularly perturbed Choquard type equation
$\mathrm{}{\mathit{\epsilon}}^{\mathrm{2}}\mathrm{\Delta}\mathit{u}\mathrm{+}\mathit{V}\mathrm{\left(}\mathit{x}\mathrm{\right)}\mathit{u}\mathrm{=}{\mathit{\epsilon}}^{\mathit{\mu}\mathrm{}\mathrm{2}}\left[\frac{\mathrm{1}}{\mathrm{}\mathit{x}{\mathrm{}}^{\mathit{\mu}}}\mathrm{\ast}(\mathit{P}\mathrm{\left(}\mathit{x}\mathrm{\right)}\mathit{G}\mathrm{\left(}\mathit{u}\mathrm{\right)})\right]\mathit{P}\mathrm{\left(}\mathit{x}\mathrm{\right)}\mathit{g}\mathrm{\left(}\mathit{u}\mathrm{\right)}$ in ℝ^{2}, where ε is a positive parameter, $\frac{\mathrm{1}}{\mathrm{}\mathit{x}{\mathrm{}}^{\mathit{\mu}}}$ with 0 < μ < 2 is the Riesz potential, ∗ is the convolution operator, V(x), P(x) are two continuous real functions and G(s) is the primitive function of g(s). Suppose that the nonlinearity g is of critical exponential growth in ℝ^{2} in the sense of the TrudingerMoser inequality, we establish some existence and concentration results of the semiclassical solutions of the Choquard type equation in the whole plane. As a particular case, the concentration appears at the maximum point set of P(x) if V(x) is a constant.
Mathematics Subject Classification: 35J25 / 35J20 / 35J60
Key words: Choquard equation / semiclassical solutions / TrudingerMoser inequality / critical exponential growth
© EDP Sciences, SMAI 2018
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