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Journal of Convex Analysis 24 (2017), No. 4, 1197--1215
Copyright Heldermann Verlag 2017

Existence of Solutions for a Nonlocal Variational Problem in R2 with Exponential Critical Growth

Claudianor O. Alves
Universidade Federal de Campina Grande, Unidade Acadêmica de Matemática, CEP: 58429-900, Campina Grande - Pb, Brazil

Minbo Yang
Dept. of Mathematics, Zhejiang Normal University, Jinhua, Zhejiang 321004, P. R. China


We study the existence of nontrivial solutions for the following class of nonlocal problem, $$ -\Delta u +V(x)u =\Big( I_\mu\ast F(x,u)\Big)f(x,u) \quad \mbox{in} \quad \mathbb{R}^2, $$ where $V$ is a positive periodic potential, $I_\mu=\frac{1}{|x|^\mu}$, $0<\mu<2$ and $F(x,s)$ is the primitive function of $f(x,s)$ in the variable $s$. By assuming that the nonlinearity $f(x,s)$ has an exponential critical growth at infinity, we prove the existence of solutions by variational methods.

Keywords: Nonlocal nonlinearities, exponential critical growth, ground state solution.

MSC: 35J50, 35J60, 35A15

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