Ground state solutions of Poho\v{z}aev type for
Kirchhoff type problems with general convolution nonlinearity and
variable potential
Abstract
This paper is devoted to dealing with the following nonlinear Kirchhoff
type problem with general convolution nonlinearity and variable
potential:
$$\left\{\begin{array}{ll}
-({a + b\int_{{\R^3}} |
\nabla u{|^2}dx})\Delta u +
V(x)u =(I_{\alpha}\ast
F(u))f(u),\quad
\text{in}\ \
\R^3,\\ u
\in H^{1} (\R^{3}),
\end{array}\right.$$ where
$a>0$, $b\geq0$ are constants,
$V\in
C^1(\R^3,[0,+\infty))$,
$f\in C(\R,\R)$,
$F(t)=\int_{0}^{t}f(s)ds$,
$I_{\alpha}:\R^{3}\rightarrow
\R$ is the Riesz potential,
$\alpha\in(0,3)$. By applying some new
analytical tricks introduced by [X.H. Tang, S.T. Chen, Adv. Nonlinear
Anal. 9 (2020) 413-437], the existence results of ground state
solutions of Poho\v{z}aev type for the above Kirchhoff
type problem are obtained under some mild assumptions on $V$ and the
general “Berestycki-Lions assumptions” on the nonlinearity $f$. Our
results generalize and improve the ones in [P. Chen, X.C. Liu, J.
Math. Anal. Appl. 473 (2019) 587-608.] and other related results in
the literature.
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