In this paper, we consider the existence of ground-state solutions to nonlinear Kirchhoff-type transmission problems by using the methods from (Silvia Cingolani and Tobias Weth, , Ann. Inst. H. Poincaré Anal. Non Linéaire (2016), no. 1, 169-197). Here, we avoid the conditions under which the nonlinear terms of $f$ and $g$ are forms of $C^1$. In particular, when $N=2$, the existence of ground-state solutions is established to the Kirchhoff-type transmission problem with exponent-type nonlinearity.
"Ground-state solutions to a Kirchhoff-type transmission problem." Topol. Methods Nonlinear Anal. 57 (1) 201 - 219, 2021. https://doi.org/10.12775/TMNA.2020.019