Existence of solutions for a class of nonlinear Choquard equations with critical growth

ABSTRACT

In this paper, we consider the nonlinear Choquard equation of the form $-\mathrm{\Delta }u+u=(I_{\alpha }\ast |u{|}^p)|u{|}^{p-2}u+|u{|}^{q-2}u\text{in}{\mathbb{R}}^N,$ where $I_{\alpha }$ is a Riesz potential, $\alpha \in (0,N)$, $N⩾4$, $p=\frac{N+\alpha }{N-2}$, $2<q<2^{\ast }$. We show the existence of a nontrivial solution of the equation. Moreover, we consider the corresponding minimizing problem and obtain a nonnegative minimizer.

KEYWORDS:

• Nonlinear Choquard equations
• Nehari manifold
• variational methods
• Schwarz rearrangement

MATHEMATICS SUBJECT CLASSIFICATION 2010:

• 35Q55
• 35R09
• 35J91

Disclosure statement

No potential conflict of interest was reported by the author.