Existence of sign-changing solution with least energy for a class of Kirchhoff-type equation in    ℝ N

Xianzhong Yao,Chunlai Mu

doi:10.14232/ejqtde.2017.1.32

Abstract

Existence of sign-changing solution with least energy for a class of Kirchhoff-type equation in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>ℝ</mml:mi> <mml:mi>N</mml:mi> </mml:msup> </mml:mrow> </mml:math>

Highlights

In this paper, we are concerned with the following Kirchhoff-type elliptic problem with general nonlinearity:a + λ |∇u|2dx + λb u2dx [−∆u + bu] = f (u), x ∈ RN, RN (1.1)where a, b > 0 are constants, λ > 0 is a parameter and N ≥ 3
We discover that researchers usually need suppose that f satisfies (H4) and (H3) or (H3), which ensure the boundedness of a minimum sequence for the corresponding functional of the Kirchhoff-type problem
To the best authors’ knowledge, there is no result on the existence of least energy sign-changing solution to Kirchhoff-type problem with nonlinearity f satisfying the hypotheses ( f3) and ( f4)

Summary

Introduction

We are concerned with the following Kirchhoff-type elliptic problem with general nonlinearity:. There has been increasing attention to the existence of sign-changing (nodal) solutions to Kirchhoff-type problem. As well it guarantees that the nodal Nehari manifold of corresponding functional of the Kirchhoff-type problem is not empty Their results can be derived by usual variational methods and quantitative deformation lemma. To the best authors’ knowledge, there is no result on the existence of least energy sign-changing (nodal) solution to Kirchhoff-type problem with nonlinearity f satisfying the hypotheses ( f3) and ( f4). We need first to introduce the energy functional for corresponding Kirchhoff-type problem (1.1) and nodal Nehari manifold. There exists a positive Λ such that, for any λ ∈ (0, Λ), the problem (1.1) have a ground state solution uλ which is constant sign and a least energy sign-changing solution vλ satisfying cλ = Jλ(vλ) > Jλ(uλ) = cλ > 0.

Preliminaries

Proof of main results