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.

This paper studies a new kind of generalized Caputo fractional differential quasi-variational inequalities in Hilbert spaces. We prove uniqueness and stability of the abstract inequality by using generalized singular Gronwall's lemma, projection operators and contraction principle. Finally, an example is given to illustrate the abstract results.