Ground state solution for a class of magnetic equation with general convolution nonlinearity

Abstract

In this paper, we consider the following magnetic Laplace nonlinear Choquard equation $ \begin{equation*} -\Delta_A u+V(x)u = (I_{\alpha}*F(|u|))\frac{f(|u|)}{|u|}u, \, \, \text{in}\, \, \mathbb{R}^N, \ \end{equation*} $ where $ u: \mathbb{R}^N\rightarrow C $, $ A: \mathbb{R}^N\rightarrow \mathbb{R}^N $ is a vector potential, $ N\geq 3 $, $ \alpha\, \in\, (N-2, N) $, $ V:\, \mathbb{R}^N \rightarrow \mathbb{R} $ is a scalar potential function and $ I_{\alpha} $ is a Riesz potential of order $ \alpha\, \in\, (N-2, N) $. Under certain assumptions on $ A(x) $, $ V(x) $ and $ f(t) $, we prove that the equation has at least a ground state solution by variational methods.

Keywords

• magnetic laplace operator
• ground state solution
• nehari manifold