Geodesic
Existence of Solutions for a Nonlocal Variational Problem in R2 with Exponential Critical Growth
Journal of convex analysis, Tome 24 (2017) no. 4, pp. 1197-1215
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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, $I_\mu=\frac{1}{|x|^\mu}$, $0\mu2$ 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.
Classification : 35J50, 35J60, 35A15
Mots-clés : Nonlocal nonlinearities, exponential critical growth, ground state solution 
@article{JCA_2017_24_4_JCA_2017_24_4_a6,
     author = {C. O. Alves and M. Yang},
     title = {Existence of {Solutions} for a {Nonlocal} {Variational} {Problem} in {R\protect\textsuperscript{2}} with {Exponential} {Critical} {Growth}},
     journal = {Journal of convex analysis},
     pages = {1197--1215},
     year = {2017},
     volume = {24},
     number = {4},
     url = {http://geodesic.mathdoc.fr/item/JCA_2017_24_4_JCA_2017_24_4_a6/}
}
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C. O. Alves; M. Yang. Existence of Solutions for a Nonlocal Variational Problem in R2 with Exponential Critical Growth. Journal of convex analysis, Tome 24 (2017) no. 4, pp. 1197-1215. http://geodesic.mathdoc.fr/item/JCA_2017_24_4_JCA_2017_24_4_a6/