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

Abstract

AbstractThis paper is dedicated to studying the following Kirchhoff-type problem: $$ \textstyle\begin{cases} -m ( \Vert \nabla u \Vert ^{2}_{L^{2}(\mathbb{R} ^{N})} )\Delta u+V(x)u=f(u), & x\in \mathbb{R} ^{N}; \\ u\in H^{1}(\mathbb{R} ^{N}), \end{cases} $$ { − m ( ∥ ∇ u ∥ L 2 ( R N ) 2 ) Δ u + V ( x ) u = f ( u ) , x ∈ R N ; u ∈ H 1 ( R N ) , where $N=1,2$ N = 1 , 2 , $m:[0,\infty )\rightarrow (0,\infty )$ m : [ 0 , ∞ ) → ( 0 , ∞ ) is a continuous function, $V:\mathbb{R} ^{N}\rightarrow \mathbb{R} $ V : R N → R is differentiable, and $f\in \mathcal{C}(\mathbb{R} ,\mathbb{R} )$ f ∈ C ( R , 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 $m(\|\nabla u\|^{2}_{L^{2}(\mathbb{R} ^{N})})$ m ( ∥ ∇ u ∥ L 2 ( R N ) 2 ) 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.