Fractional p(·)-Kirchhoff Type Problems Involving Variable Exponent Logarithmic Nonlinearity

Jiabin Zuo,Debajyoti Choudhuri,Amita Soni

doi:10.3390/fractalfract6020106

Abstract

In this paper, we investigate a fractional p(·)-Kirchhoff type problem involving variable exponent logarithmic nonlinearity. With the help of the Nehari manifold approach, the existence and multiplicity of nontrivial weak solutions for the above problem are obtained. The main aspect and challenges of this paper are the presence of double non-local terms and logarithmic nonlinearity.

Highlights

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We point out that Xiang et al [26] investigated the existence of two local least energy solutions for fractional pKirchhoff problems involving logarithmic nonlinearity by means of the Nehari manifold approach
First of all, we review some basic properties about the variable exponent

Summary

Introduction

Many mathematicians were concerned with equations involving the operator and studied it extensively, see [12,13,14,15,16,17] This combination of fractional p( x )-Laplace operators and Kirchhoff functions is very interesting. We point out that Xiang et al [26] investigated the existence of two local least energy solutions for fractional pKirchhoff problems involving logarithmic nonlinearity by means of the Nehari manifold approach This method is used essentially because the functional corresponding to the equation is not bounded below in the whole workspace, so it is difficult to find the critical points in the whole workspace, and we need to find the critical points on a smaller set.

Functional Analytic Setup

The Proof of Result

Conclusions