Ground state for Choquard equation with doubly critical growth nonlinearity

Fuyi Li,Lei Long,Yongyan Huang,Zhanping Liang

doi:10.14232/ejqtde.2019.1.33

Abstract

Ground state for Choquard equation with doubly critical growth nonlinearity

Highlights

In view of the Hardy–Littlewood–Sobolev inequality, see Lemma 2.1 below, it can be shown that the energy functional corresponding to (1.2), for every α ∈ (0, N), is well defined on
The solution u obtained in Theorem 1.1 is a ground state solution in the sense that it minimizes the corresponding energy functional J among all nontrivial solutions
The proof of Theorem 1.2 relies on two ingredients: the nontrivial nature of solution to the equation (1.3) up to translation under the strict inequality c < min 1 p1/(p−1), 1 2

Summary

Introduction

X ∈ RN \ {0}, Γ denotes the Gamma function, F(t) = |t|p/p + |t|q/q, f (t) = |t|p−2t + |t|q−2t for all t ∈ R, and the potential function V ∈ C(RN) and satisfies (V) there exist V0, V∞ > 0 such that V0 V(x) V∞ for all x ∈ RN, and lim|x|→∞ V(x) = V∞. The existence of ground states of equation (1.2) was obtained in [6, 8, 9] by variational methods. In view of the Hardy–Littlewood–Sobolev inequality, see Lemma 2.1 below, it can be shown that the energy functional corresponding to (1.2), for every α ∈ (0, N), is well defined on

Results

Conclusion