Nonlocal Hénon type problem with nonlinearities involving slightly subcritical growth

IF 2.4 2区 数学 Q1 MATHEMATICS
Imene Bendahou , Zied Khemiri , Fethi Mahmoudi
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引用次数: 0

Abstract

In this paper, we study the existence of solutions for the following nonlocal superlinear elliptic problem(0.1){(Δ)su=β(x)upεin Ω,u=0in RnΩ, where s(0,1),n>2s,p:=n+2sn2s is the Sobolev critical exponent, ΩRn is a smooth bounded domain with Lipschitz boundary, (Δ)s is the fractional Laplace operator and βC2(Ω) is a bounded positive continuous function. We assume that there exists a nondegenerate critical point ξΩ of the restriction of β to the boundary ∂Ω such that(β(ξ)n2s2s)η(ξ)>0. Given any integer k1, we show that for ε>0 small enough, problem (0.1) has a positive solution, which is a sum of k bubbles which concentrate at ξ as ε tends to zero. Also, we prove the existence of nodal (sign changing) solution whose shape resembles a sum of a positive bubble and a negative bubble near the point ξ. This work can be seen as a nonlocal analogue of the result by Dávila, Faya and Mahmoudi, see [28].

涉及轻微次临界增长的非线性非局部赫农型问题
本文研究下列非局部超线性椭圆问题(0.1){(-Δ)su=β(x)up-εin Ω,u=0in Rn﹨Ω,其中 s∈(0,1),n>2s,p:=n+2sn-2s 是索博列夫临界指数,Ω⊂Rn 是具有 Lipschitz 边界的光滑有界域,(-Δ)s 是分数拉普拉斯算子,β∈C2(Ω‾) 是有界正连续函数。我们假定存在一个非enerate 临界点ξ⁎∈∂Ω,使得∇(β(ξ⁎)-n-2s2s)⋅η(ξ⁎)>0。给定任意整数 k≥1,我们证明,对于足够小的ε>0,问题 (0.1) 有一个正解,它是 k 个气泡之和,当 ε 趋近于零时,这些气泡集中于 ξ⁎。此外,我们还证明了节点(符号变化)解的存在,其形状类似于ξ⁎点附近的一个正气泡和一个负气泡之和。这项工作可以看作是 Dávila、Faya 和 Mahmoudi 成果的非局部类比,见 [28]。
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来源期刊
CiteScore
4.40
自引率
8.30%
发文量
543
审稿时长
9 months
期刊介绍: The Journal of Differential Equations is concerned with the theory and the application of differential equations. The articles published are addressed not only to mathematicians but also to those engineers, physicists, and other scientists for whom differential equations are valuable research tools. Research Areas Include: • Mathematical control theory • Ordinary differential equations • Partial differential equations • Stochastic differential equations • Topological dynamics • Related topics
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