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

Abstract

Abstract In this paper, we consider the following nonlinear problem with general nonlinearity and nonlocal convolution term: $$ \textstyle\begin{cases} -\Delta u+V(x)u+(I_{\alpha }\ast \vert u \vert ^{q}) \vert u \vert ^{q-2}u=f(u), \quad x\in {\mathbb{R}}^{3}, \\ u\in H^{1}(\mathbb{R}^{3}), \quad \end{cases} $$ { − Δ u + V ( x ) u + ( I α ∗ | u | q ) | u | q − 2 u = f ( u ) , x ∈ R 3 , u ∈ H 1 ( R 3 ) , where $a\in (0,3)$ a ∈ ( 0 , 3 ) , $q\in [1+\frac{\alpha }{3},3+\alpha )$ q ∈ [ 1 + α 3 , 3 + α ) , $I_{\alpha }:\mathbb{R}^{3}\rightarrow \mathbb{R}$ I α : R 3 → R is the Riesz potential, $V\in \mathcal{C}(\mathbb{R}^{3},[0,\infty ))$ V ∈ C ( R 3 , [ 0 , ∞ ) ) , $f\in \mathcal{C}(\mathbb{R},\mathbb{R})$ f ∈ C ( R , R ) and $F(t)=\int _{0}^{t}f(s)\,ds$ F ( t ) = ∫ 0 t f ( s ) d s satisfies $\lim_{|t|\to \infty }F(t)/|t|^{\sigma }=\infty $ lim | t | → ∞ F ( t ) / | t | σ = ∞ with $\sigma =\min \{2,\frac{2\beta +2}{\beta }\}$ σ = min { 2 , 2 β + 2 β } where $\beta =\frac{ \alpha +2}{2(q-1)}$ 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.