Existence of multiple positive solutions for nonhomogeneous fractional Laplace problems with critical growth

Fang Wang,Yajing Zhang

doi:10.1186/s13661-019-1287-9

Abstract

We prove the existence of multiple positive solutions of fractional Laplace problems with critical growth by using the method of monotonic iteration and variational methods.

Highlights

Considerable attention has been devoted to fractional and non-local operators of elliptic type in recent years, both for their interesting theoretical structure and in view of concrete applications, like flame propagation, chemical reactions of liquids, population dynamics, geophysical fluid dynamics, and American options; see [3, 7, 19, 20] and the references therein.In this paper we consider the following critical problem: ⎧⎨(– )su = λu + |u|p–2u + γ g(x) in Ω,(P)γ ⎩u = 0 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 :=, g
We prove Theorem 1.1 by the method of monotonic iteration, known as the super and subsolution method, which is a basic tool in nonlinear partial differential equations
2 Existence of the first positive solution we prove existence of the first solution of (P)γ by the method of monotonic iteration

Summary

Introduction

Theorem 1.2 For λ ∈ [0, λ1), γ ∈ (0, γ ∗), where γ ∗ is the one in Theorem 1.1, problem (P)γ admits at least two positive solutions. 2 we prove the existence of the first solution of (P)γ by the method of monotonic iteration. 3 we prove the existence of the second solution of (P)γ by variational methods. Lemma 2.1 For λ ∈ [0, λ1) there exists a constant γ > 0 such that (P)γ has no positive solution for γ > γ .

Results

Conclusion