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

中文翻译：

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低维一般非线性的Kirchhoff型方程的基态解

本文致力于研究以下Kirchhoff型问题：$$ \ textstyle \ begin {cases -m（\ Vert \ nabla u \ Vert ^ {2} _ {L ^ {2}（\ mathbb {R} ^ {N}）}）\ Delta u + V（x）u = f（u），和x \ in \ mathbb {R} ^ {N}; \\ u \ in H ^ {1}（\ mathbb {R} ^ {N}），\ end {cases} $$，其中$ N = 1,2 $，$ m：[0，\ infty）\ rightarrow（ 0，\ infty）$是连续函数，$ V：\ mathbb {R} ^ {N} \ rightarrow \ mathbb {R} $是可微的，而$ f \ in \ mathcal {C}（\ mathbb {R} ，\ mathbb {R}）$。在某些关于V，m和f的假设下，我们获得了Nehari–Pohozaev类型的基态解和最小能量解的存在。特别是，存在非局部项$ m（\ | \ nabla u \ | ^ {2} _ {L ^ {2}（\ mathbb {R} ^ {N}）}）$和Hardy不等式和Sobolev's不存在低维不等式使问题更加复杂。为了克服上述困难，