Multiplicity of Positive Solutions for a Nonlocal Elliptic Problem Involving Critical Sobolev-Hardy Exponents and Concav
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Wuhan Institute Physics and Mathematics, Chinese Academy of Sciences, 2020
http://actams.wipm.ac.cn
MULTIPLICITY OF POSITIVE SOLUTIONS FOR A NONLOCAL ELLIPTIC PROBLEM INVOLVING CRITICAL SOBOLEV-HARDY EXPONENTS AND CONCAVE-CONVEX NONLINEARITIES∗
Ü7I)
Jinguo ZHANG (
School of Mathematics, Jiangxi Normal University, Nanchang 330022, China E-mail : [email protected]
Nì)
Tsing-San HSU (
†
Center for General Education, Chang Gung University, Tao-Yuan, Taiwan, China E-mail : [email protected] Abstract In this article, we study the following critical problem involving the fractional Laplacian: ∗ q−2 |u|2α (t)−2 u (−∆) α2 u − γ u = λ |u| + in Ω, |x|α |x|s |x|t u=0 in RN \Ω,
where Ω ⊂ RN (N > α) is a bounded smooth domain containing the origin, α ∈ (0, 2), 0 ≤ s, t < α, 1 ≤ q < 2, λ > 0, 2∗α (t) = 2(N−t) is the fractional critical Sobolev-Hardy N−α exponent, 0 ≤ γ < γH , and γH is the sharp constant of the Sobolev-Hardy inequality. We deal with the existence of multiple solutions for the above problem by means of variational methods and analytic techniques. Key words
Fractional Laplacian; Hardy potential; multiple positive solutions; critical Sobolev-Hardy exponent
2010 MR Subject Classification
1
47G20; 35J50; 35B09
Introduction
In this article, we are concerned with the existence and multiplicity of positive solutions for the following nonlocal elliptic problem with Hardy potential: q−2 2∗ α (t)−2 u (−∆) α2 u − γ u = λ |u| u + |u| in Ω, |x|α |x|s |x|t (1.1) N u=0 in R \Ω, α
where (−∆) 2 is the fractional Laplace operator with α ∈ (0, 2), Ω is a smooth bounded domain −t) in RN containing 0 in its interior, 0 ≤ s , t < α, 1 ≤ q < 2, λ > 0, 2∗α (t) = 2(N N −α is the ∗ Received
November 7, 2018. author
† Corresponding
680
ACTA MATHEMATICA SCIENTIA
Vol.40 Ser.B Γ2 ( N +α )
4 so-called fractional critical Sobolev-Hardy exponent, and 0 ≤ γ < γH , where γH = 2α Γ2 ( N −α 4 ) is the best fractional Hardy inequality: Z Z α |u|2 1 α dx ≤ |(−∆) 4 u|2 dx, u ∈ X02 (Ω). α γH RN Ω |x| α
Here, X02 (Ω) is a Hilbert space, and it is defined as o n α α X02 (Ω) = u ∈ H 2 (RN ) | u = 0 a.e. in RN \Ω , and endowed with the following norm Z α kuk 2 = c(N, α) X0 (Ω)
RN
Z
RN
12 |u(x) − u(y)|2 dxdy . |x − y|N +α α
α
It is well-known that, if α = 2, the operator (−∆) 2 is defined as (−∆) 2 u = −∆u =
N P
i=1
which is a local operator. The general problem u − ∆u − γ α = f (x, u) |x| u=0
∂2 u , ∂x2i
in Ω, (1.2) on ∂Ω
has been studied extensively. In case of γ = 0, Bahri-Coron [1] obtained the existence and ∗ multiplicity results for solutions of problem (1.2) with f (x, u) = |u|2 −2 u, and Passaseo [21] proved that problem (1.2) admits at least two weak solutions in H01 (Ω). In the case of γ 6= 0, we recall that the existence of solutions for problem (1.1) has been obtained under different hypotheses on f (x, u); see Cao-Han [6], Chen [5], A. Ferrero, F. Gazzola [15], Smets [22], Terracini [27], Cao-Kang [8], Filippucci et al [17], and the references therein. In this article, our ma
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