Ground state solutions for Kirchhoff-type equations with general nonlinearity in low dimension

Abstract This paper is dedicated to studying the following Kirchhoff-type problem: $$ \textstyle\begin{cases} -m ( \Vert \nabla u ^{2}_{L^{2}(\mathbb{R} ^{N})} )\Delta u+V(x)u=f(u), & x\in \mathbb{R} ^{N}; \\ u\in H^{1}(\mathbb{R} ^{N}), \end{cases} { ? m ( ? ? u L 2 R N ) ? + V x = f , ? ; H 1 where $N=1,2$ , $m:[0,\infty )\rightarrow (0,\infty )$ : [ 0 ? ? a continuous function, $V:\mathbb{R} ^{N}\rightarrow $ differentiable, and $f\in \mathcal{C}(\mathbb{R} ,\mathbb{R} C . We obtain existence of ground state solution Nehari–Pohozaev type least energy under some assumptions on V m f Especially, nonlocal term $m(\|\nabla u\|^{2}_{L^{2}(\mathbb{R} ^{N})})$ lack Hardy’s inequality Sobolev’s in low dimension make problem more complicated. To overcome above-mentioned difficulties, new inequalities subtle analyses are introduced.