MULTIPLE SOLUTIONS FOR A FRACTIONAL LAPLACIAN SYSTEM INVOLVING CRITICAL SOBOLEV-HARDY EXPONENTS AND HOMOGENEOUS TERM

Jinguo Zhang,Tsing-San Hsu

doi:10.3846/mma.2020.7704

Abstract

In this paper, we deal with a class of fractional Laplacian system with critical Sobolev-Hardy exponents and sign-changing weight functions in a bounded domain. By exploiting the Nehari manifold and variational methods, some new existence and multiplicity results are obtain.

Highlights

We mainly study the following system of fractional elliptic equations:
Using the variational method and Nehari manifold method, they found that the problem (1.3) has at least two positive solutions if the parameters λ, μ > 0 satisfied a certain condition
Motivated by [20], in this paper we focus on the general case f, g possibly change sign in Ω and F positively 2∗s(β)-homogeneous, we shall complement the results of [18, 20] and extend the results of [11, 12, 13, 14] to the fractional Laplacian operator

Summary

Introduction

We mainly study the following system of fractional elliptic equations:. Using the variational method and Nehari manifold method, they found that the problem (1.3) has at least two positive solutions if the parameters λ, μ > 0 satisfied a certain condition. Problems (1.2) and (1.3) aroused the interesting results due to the lack of compactness for involving the critical exponent; the associated energy functionals do not satisfy the Palais-Smale condition in general. We show that the system (1.1) has at least two positive solutions when the parameters λ, μ and weight functions f , g satisfied some certain conditions. It should be mentioned that in [8, 9, 10, 15, 22], some problems involving fractional Laplacian operator were investigated by the Nehari manifold and fibering method. To the best of our knowledge, the results are new for the critical fractional Laplacian problem with Hardy potential and homogeneous term. Means on(1) → 0 as n → ∞; The dual space of W will be denoted by W −1; C, Ci will denote various positive constants which may vary from line to line

Preliminaries

The Palais-Smale condition

Nehari manifold

Findings

Proof of Theorem 2