Least energy sign-changing solutions for fourth-order Kirchhoff-type equation with potential vanishing at infinity
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Least energy sign-changing solutions for fourth-order Kirchhoff-type equation with potential vanishing at infinity Hua-Bo Zhang1 · Wen Guan1 Received: 23 September 2019 © Korean Society for Informatics and Computational Applied Mathematics 2020
Abstract In this paper, we study the following fourth-order Kirchhoff-type equation |∇u|2 d x u + V (x)u = K (x) f (u), x in R N , 2 u − a + b RN
with the potential V (x) vanishing at infinity. Under suitable conditions, by using constraint variational method and the quantitative deformation lemma, we obtain a least energy sign-changing (or nodal) solution to this problem. Moreover, we prove that this least energy sign-changing solution has precisely two nodal domains. Keywords Fourth-order Kirchhoff-type equation · Nonlocal term · Variation methods · Sign-changing solutions Mathematics Subject Classification 35J60 · 35J20
1 Introduction and main result In this article, we are interested in the existence of the least energy sign-changing solution for the following fourth-order Kirchhoff-type equation |∇u|2 d x u + V (x)u = K (x) f (u), x ∈ R N , (1.1) 2 u − a + b RN
where 5 ≤ N ≤ 7, 2 denotes the biharmonic operator, ∇u is the spatial gradient of u, and a, b > 0 are real constants. Throughout this paper, we say that (V , K ) ∈ K if the following conditions hold: (V K 0 ) V (x), K (x) > 0 for all x ∈ R N and K ∈ L ∞ (R N ).
B 1
Wen Guan [email protected] Department of Applied Mathematics, Lanzhou University of Technology, Lanzhou 730050, Gansu, People’s Republic of China
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H.-B. Zhang, W. Guan
(V K 1 ) If {An }n ⊂ R N is a sequence of Borel sets such that |An | ≤ R, for all n ∈ N and some R > 0, then lim K (x) = 0, uniformly in n ∈ N. r →∞
An
Brc (0)
One of the following conditions occurs: (V K 2 ) K /V ∈ L ∞ (R N ); or (V K 3 ) there is p ∈ (2, 2∗ ) such that K (x) 2∗ − p
|V (x)| 2∗ −2
→ 0 as |x| → ∞, where 2∗ =
( f1)
2N is the critical Sobolev exponent. N −4
As for the function f , we assume f ∈ C 1 (R, R) and satisfies: f (t) = o(|t|) as t → 0 if (V K 2 ) holds.
( f2 ) lim sup t→0
f (t) < ∞, if (V K 3 ) holds. |t| p−1
( f 3 ) f has a subcritical growth, that is, lim sup |t|→∞
f (t) = 0. |t|2∗ −1
t F(t) 4 = ∞, where F(t) = 0 f (s)ds. t |t|→∞ f (t) is an increasing function of t ∈ R\{0}. |t|3
( f 4 ) lim ( f5)
The motivation to study problem (1.1) comes from Kirchhoff equations of the type 2 u − (a + b |∇u|2 d x)u = f (x, u), x ∈ , (1.2)
where ⊂ R N is a bounded domain, a > 0, b > 0 and u satisfies some boundary conditions. The problem (1.2) is related to the following stationary analogue of the equation of Kirchhoff type |∇u|2 d x)u = f (x, u), x ∈ , (1.3) u tt − 2 u − (a + b
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Least energy sign-changing solutions for fourth-order…
which is used to describe some phenomena appeared in different physical, engineering and other sciences because it is regarded as a good approximation for describing nonlinear vibrations of beams or plates, see [4,8]. Problems (1.1) and (1.2) are nonlocal problems because t
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