This paper is dedicated to studying the following Kirchhoff-type problem $$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} -\left( a+b\int _{\mathbb {R}^3}|\nabla u|^2\mathrm {d}x\right) \triangle u+V(x)u=f(u), &{} x\in \mathbb {R}^3; \\ u\in H^1(\mathbb {R}^3), \end{array} \right. \end{aligned}$$ - a + b ∫ R 3 | ∇ u | 2 d x ▵ u + V ( x ) u = f ( u ) , x ∈ R 3 ; u ∈ H 1 ( R 3 ) , where $$a>0,\,b\ge 0$$ a > 0 , b ≥ 0 are two constants, V(x) is differentiable and $$f\in \mathcal {C}(\mathbb {R}, \mathbb {R})$$ f ∈ C ( R , R ) . By introducing some new tricks, we prove that the above problem admits a ground state solution of Nehari–Pohozaev type and a least energy solution under some mild assumptions on V and f. Our results generalize and improve the ones in Guo (J Differ Equ 259:2884–2902, 2015) and Li and Ye (J Differ Equ 257:566–600, 2014) and some other related literature.
Calculus of Variations and Partial Differential Equations – Springer Journals
Published: Jul 10, 2017
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