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

• Published in: Calculus of variations and partial differential equations Vol. 56; no. 4; pp. 1 - 25
• Main Authors: Tang, X. H., Chen, Sitong
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01-08-2017; Springer Nature B.V
• Subjects: Analysis; Calculus of Variations and Optimal Control; Optimization; Control; Ground state; Mathematical and Computational Physics; Mathematics; Mathematics and Statistics; Systems Theory; Theoretical; 35J65; 35J20
• ISSN: 0944-2669; 1432-0835
• DOI: 10.1007/s00526-017-1214-9

This paper is dedicated to studying the following Kirchhoff-type problem
0.1
-
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
≥
0
are two constants,
V
(
x
) is differentiable and
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.