Existence of multiple positive solutions for fractional Laplace problems with critical growth and sign-changing weight in non-contractible domains

Lu Pang,Xueqin Li,Yajing Zhang

doi:10.1186/s13661-019-1193-1

Abstract

We prove the existence of multiple positive solutions for a fractional Laplace problem with critical growth and sign-changing weight in non-contractible domains.

Highlights

In this paper we consider the following critical problem involving fractional Laplacian:⎧ ⎪⎪⎨(– )su = a(x)up–1 + u2∗s –1 ⎪⎪⎩uu > =0 0 in Ω, in Ω, in RN \ Ω, (1.1)where s ∈ (0, 1) is fixed and (– )s is the fractional Laplace operator, Ω ⊂ RN (N > 2s) is a smooth bounded domain, < p 2∗s :=
We prove the existence of multiple positive solutions for a fractional Laplace problem with critical growth and sign-changing weight in non-contractible domains
Proof By Lemma 5.10, there exists ε∗ ∈ (0, ε0) such that, for ε ∈ (0, ε∗), uε,e ∈ Σ and u2(1–θ )ε,e for all e

Summary

Introduction

Many papers have studied the existence and multiplicity of positive solutions of the problem similar to (1.2), see [16, 18, 37, 39]. Proof Assume by contradiction that there exists a ∈ C(Ω ) with |a+|q < σ1 such that N = ∅. Lemma 3.1 (i) m+ < 0 if function a satisfies |a+|q ∈ (0, σ1); (ii) there exists positive constant c0 such that m– ≥ c0 if |a+|q < σ2.

Results

Conclusion