Ground state solutions of Nehari–Pohožaev type for a kind of nonlinear problem with general nonlinearity and nonlocal convolution term

Abstract

In this paper, we consider the following nonlinear problem with general nonlinearity and nonlocal convolution term:-u + V(x)u + (I α * |u| q)|u| q-2 u = f (u), x ∈ R 3 , u ∈ H 1 (R 3), where a ∈ (0, 3), q ∈ [1 + α 3 , 3 + α), I α : R 3 → R is the Riesz potential, V ∈ C(R 3 , [0, ∞)), f ∈ C(R, R) and F(t) = t 0 f (s) ds satisfies lim |t|→∞ F(t)/|t| σ = ∞ with σ = min{2, 2β+2 β } where β = α+2 2(q-1). By using new analytic techniques and new inequalities, we prove the above system admits a ground state solution under mild assumptions on V and f .