# Ground state solutions of Nehari–Pohozaev type for Kirchhoff-type problems with general potentials

Ground state solutions of Nehari–Pohozaev type for Kirchhoff-type problems with general potentials 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Calculus of Variations and Partial Differential Equations Springer Journals

# Ground state solutions of Nehari–Pohozaev type for Kirchhoff-type problems with general potentials

, Volume 56 (4) – Jul 10, 2017
25 pages

/lp/springer_journal/ground-state-solutions-of-nehari-pohozaev-type-for-kirchhoff-type-DMwKb7shvI
Publisher
Springer Berlin Heidelberg
Copyright © 2017 by Springer-Verlag GmbH Germany
Subject
Mathematics; Analysis; Systems Theory, Control; Calculus of Variations and Optimal Control; Optimization; Theoretical, Mathematical and Computational Physics
ISSN
0944-2669
eISSN
1432-0835
D.O.I.
10.1007/s00526-017-1214-9
Publisher site
See Article on Publisher Site

### Abstract

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.

### Journal

Calculus of Variations and Partial Differential EquationsSpringer Journals

Published: Jul 10, 2017

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