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Existence and Regularity of Solutions for a Choquard Equation with Zero Mass

  • Claudianor O. Alves
  • Jianfu Yang
Article
  • 2 Downloads

Abstract

This paper concerns with the existence and regularity of solutions for the following Choquard type equation,
$$-\Delta_u = \big(I_{\mu} * F(u)\big) f(u) {\rm in} \mathbb{R}^3, \quad \quad (P)$$
where \({I_\mu = \frac{1}{|x|^\mu}, 0 < \mu < 3}\), is the Riesz potential, \({F(s)}\) is the primitive of the continuous function f(s), and \({I_{\mu} * F(u)}\) denotes the convolution of \({I_{\mu}}\) and F(u). By using the variational method, we prove that problem (P), in the zero mass case, possesses at least a nontrivial solution under certain conditions on f.

Mathematics Subject Classification (2010)

35J50 35J60 35A15 

Keywords

Choquard equation nonlocal nonlinearities zero mass existence of solution regularity 

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© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Universidade Federal de Campina Grande Unidade Acadêmica de MatemáticaCampina GrandeBrazil
  2. 2.Department of MathematicsJiangxi Normal UniversityNanchang, JiangxiP.R. China

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