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

By Luca Battaglia, Jean Van Schaftingen. 2016

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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 Iα{I_{\alpha}} is the Riesz potential of order α on the plane ℝ2{\mathbb{R}^{2}} under general nontriviality, growth and subcriticality on the nonlinearity F∈C1⁢(ℝ,ℝ){F\in C^{1}(\mathbb{R},\mathbb{R})}.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/61cbcba901c62970443dd37677e45ed3f0cd9507