Abstract
This paper is concerned with the following Kirchhoff-type problem with convolution nonlinearity:
-\bigg{(}a+b\int_{\mathbb{R}^{3}}\lvert\nabla u|^{2}\,\mathrm{d}x\bigg{)}%
\Delta u+V(x)u=(I_{\alpha}*F(u))f(u),\quad x\in{\mathbb{R}}^{3},\,u\in H^{1}(%
\mathbb{R}^{3}),
where
{a,b>0}
,
{I_{\alpha}\colon\mathbb{R}^{3}\rightarrow\mathbb{R}}
, with
{\alpha\in(0,3)}
, is the Riesz potential,
{V\in\mathcal{C}(\mathbb{R}^{3},[0,\infty))}
,
{f\in\mathcal{C}(\mathbb{R},\mathbb{R})}
and
{F(t)\kern-1.0pt=\kern-1.0pt\int_{0}^{t}f(s)\,\mathrm{d}s}
.
By using variational and some new analytical techniques, we prove that the above problem admits ground state solutions
under mild assumptions on V and f. Moreover, we give a non-existence result.
In particular, our results extend and improve the existing ones,
and fill a gap in the case where
{f(u)=|u|^{q-2}u}
, with
{q\in(1+\alpha/3,2]}
.