Existence of Groundstates for a Class of Nonlinear Choquard Equations in the Plane

Abstract

Abstract We prove the existence of a nontrivial groundstate solution for the class of nonlinear Choquard equations - Δ ⁢ u + u = ( I α * F ⁢ ( u ) ) ⁢ F ′ ⁢ ( u )   in  ⁢ ℝ 2 , -\Delta u+u=\bigl{(}I_{\alpha}*F(u)\bigr{)}F^{\prime}(u)\quad\text{in }\mathbb% {R}^{2}, where I α {I_{\alpha}} is the Riesz potential of order α on the plane ℝ 2 {\mathbb{R}^{2}} under general nontriviality, growth and subcriticality on the nonlinearity F ∈ C 1 ⁢ ( ℝ , ℝ ) {F\in C^{1}(\mathbb{R},\mathbb{R})} .