The Existence of Nontrivial Solution for a Class of Kirchhoff-Type Equation of General Convolution Nonlinearity without Any Growth Conditions

Li Zhou,Chuanxi Zhu

doi:10.3390/axioms10030163

Abstract

In this paper, we consider the following Kirchhoff-type equation: {u∈H1(RN),−(a+b∫RN|∇u|2dx)Δu+V(x)u=(Iα∗F(u))f(u)+λg(u),inRN, where a>0, b≥0, λ>0, α∈(N−2,N), N≥3, V:RN→R is a potential function and Iα is a Riesz potential of order α∈(N−2,N). Under certain assumptions on V(x), f(u) and g(u), we prove that the equation has at least one nontrivial solution by variational methods.

Highlights

Under certain assumptions on V ( x ), f (u) and g(u), we prove that the equation has at least one nontrivial solution by variational methods
We study the following Kirchhoff-type equation: R
In [10], the authors obtained the existence of a nontrivial solution for the following Kirchhoff-type equation

Summary

Introduction

We study the following Kirchhoff-type equation:. −( a + b RN |∇u|2 dx )∆u + V ( x )u = ( Iα ∗ F (u)) f (u) + λg(u), in R N , u ∈ H 1 (R N ),. We assume that the function g ∈ C (R, R) satisfies the following. Many scholars have studied the existence of nontrivial solutions for the Kirchhoff-type problem: 4.0/). In [4], Guo studied the following Kirchhoff-type problem. In [10], the authors obtained the existence of a nontrivial solution for the following Kirchhoff-type equation. In [10], there is no Ambrosetti–Rabinowtiz and no growth condition Their conclusion holds for general supercritical nonlinearity. Motivated by the works mentioned above, especially by [10,21,22], we consider the combination of the two types of Equations (4) and (5) and extend to the general convolution case in R N.

Preliminaries

Variational Formulation