Nodal Solution for a Kirchhoff-Type Problem in R<sup><i>N</i></sup>
- 1 College of Science, University of Shanghai for Science and Technology, Shanghai, China
Abstract
In this paper, we study the existence of nodal solutions of the following general Sch ö dinger-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 Sch ö dinger equations: Because the general Sch ö dinger-Kirchhoff type problem contains the nonlocal term, it implies that the equation ( KP 1) is no longer a pointwise identity and is very different from classical elliptic equations. By introducing a variable replacement, we first prove that ( KP 1) is equivalent to the following problem: where and G -1 is the inverse of G . Next, we prove that ( KP 2) is equivalent to the following system with respect to : For every integer k > 0, radial solutions of ( KP 1) 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.
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