Abstract

We study the existence of nontrivial solutions for the following class of nonlocal problem, −Δu+V(x)u=(Iμ∗F(x,u))f(x,u)inR2,-\Delta u +V(x)u =\Big( I_\mu\ast F(x,u)\Big)f(x,u) \quad \text{in} \quad \mathbb{R}^2, where VV is a positive periodic potential, Iμ=1∣x∣μI_\mu=\frac{1}{|x|^\mu}, 0<μ<20<\mu<2 and F(x,s)F(x,s) is the primitive function of f(x,s)f(x,s) in the variable ss. By assuming that the nonlinearity f(x,s)f(x,s) has an exponential critical growth at infinity, we prove the existence of solutions by variational methods.

Author information

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Claudianor O. Alves

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

coalves@dme.ufcg.edu.br

Minbo Yang

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

mbyang@zjnu.edu.cn

Keywords

  • Nonlocal nonlinearities
  • exponential critical growth
  • ground state solution

Mathematics Subject Classification

35J50, 35J60, 35A15

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Suggested citation

C. O. Alves, M. Yang. “Existence of Solutions for a Nonlocal Variational Problem in ℝ^2 with Exponential Critical Growth.” Journal of Convex Analysis 24 (2017), No. 4, 1197–1215. https://doi.org/10.68381/jca24069

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