TITLE:
Nodal Solution for a Kirchhoff-Type Problem in RN
AUTHORS:
Leilei Sha
KEYWORDS:
Sign-Changing Solutions, Nonlocal Operator, Kirchhoff-Type Equations
JOURNAL NAME:
Applied Mathematics,
Vol.11 No.1,
January
10,
2020
ABSTRACT:
In this paper, we study the existence of nodal solutions of the following general Schödinger-Kirchhoff type problem:
where a,b > 0, N ≥ 3, g : R → R+ is an even differential function and
g''(s) ≥ 0 for all s ≥ 0, h : R → R is an odd differential function. These equations are related to the generalized quasilinear Schödinger equations:
Because the general Schödinger-Kirchhoff type problem contains the nonlocal term, it implies that the equation (KP1) is no longer a pointwise identity and is very different from classical elliptic equations. By introducing a variable replacement, we first prove that (KP1) is equivalent to the following problem:
whereand G-1 is the inverse of G. Next, we prove that (KP2) is equivalent to the following system with respect to :
For every integer k > 0, radial solutions of (KP1) with exactly k nodes are obtained by dealing with the system (S) under some appropriate assumptions. Moreover, this paper established the nonexistence results if N ≥ 4 and b is sufficiently large.