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: {−Δu+V(x)u+(Iα∗|u|q)|u|q−2u=f(u),x∈R3,u∈H1(R3), $$ \textstyle\begin{cases} -\Delta u+V(x)u+(I_{\alpha }\ast \vert u \vert ^{q}) \vert u \vert ^{q-2}u=f(u), \quad x\in {\mathbb{R}}^{3}, \\ u\in H^{1}(\mathbb{R}^{3}), \quad \end{cases} $$ where a∈(0,3) $a\in (0,3)$, q∈[1+α3,3+α) $q\in [1+\frac{\alpha }{3},3+\alpha )$, Iα:R3→R $I_{\alpha }:\mathbb{R}^{3}\rightarrow \mathbb{R}$ is the Riesz potential, V∈C(R3,[0,∞)) $V\in \mathcal{C}(\mathbb{R}^{3},[0,\infty ))$, f∈C(R,R) $f\in \mathcal{C}(\mathbb{R},\mathbb{R})$ and F(t)=∫0tf(s)ds $F(t)=\int _{0}^{t}f(s)\,ds$ satisfies lim|t|→∞F(t)/|t|σ=∞ $\lim_{|t|\to \infty }F(t)/|t|^{\sigma }=\infty $ with σ=min{2,2β+2β} $\sigma =\min \{2,\frac{2\beta +2}{\beta }\}$ where β=α+22(q−1) $\beta =\frac{ \alpha +2}{2(q-1)}$. 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.

Keywords

• Schrödinger–Poisson equation
• Nehari–Pohožaev type
• Ground state solutions
• Convolution nonlinearity