The Perturbation Problem of an Elliptic System with Sobolev Critical Growth
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Wuhan Institute Physics and Mathematics, Chinese Academy of Sciences, 2020
http://actams.wipm.ac.cn
THE PERTURBATION PROBLEM OF AN ELLIPTIC SYSTEM WITH SOBOLEV CRITICAL GROWTH∗
oÛ)
Qi LI (
School of Mathematics and Statistics, Central China Normal University, Wuhan 430079, China E-mail : [email protected] Abstract exponent:
In this paper, we study the following perturbation problem with Sobolev critical ∗ α −∆u = (1 + εK(x))u2 −1 + ∗ uα−1 v β + εh(x)up , 2 ∗ β −∆v = (1 + εQ(x))v 2 −1 + ∗ uα v β−1 + εl(x)v q , 2 u > 0, v > 0,
x ∈ RN , (0.1)
x ∈ RN , x ∈ RN ,
2N where 0 < p, q < 1, α + β = 2∗ := N−2 , α, β ≥ 2 and N = 3, 4. Using a perturbation argument and a finite dimensional reduction method, we get the existence of positive solutions to problem (0.1) and the asymptotic property of the solutions.
Key words
perturbation argument; finite dimensional reduction method; critical exponent
2010 MR Subject Classification
1
35J47; 35J50; 58J37
Introduction In this paper, we consider the following coupled elliptic system: ∗ α −∆u = (1 + εK(x))u2 −1 + ∗ uα−1 v β + εh(x)up , x ∈ RN , 2 ∗ β 2 −1 −∆v = (1 + εQ(x))v x ∈ RN , + ∗ uα v β−1 + εl(x)v q , 2 u > 0, v > 0, x ∈ RN , u, v ∈ D1,2 RN ,
(1.1)
1,2 where 0 < p, q < 1, α + β = 2∗ := N2N (RN ) is the −2 , α, β ≥ 2 and N = 3, 4. Here D completion of C0∞ (RN ) with respect to the norm Z 12 kuk = |∇u|2 . RN
Systems like (1.1) are general versions of nonlinear elliptic systems, such as the BoseEinstein condensate, that arise in mathematical physics. We refer to [1–9, 21, 22] and references therein for more on systems (equations) with both critical and subcritical exponent. In ∗ Received January 16, 2019; revised September 3, 2019. Q. Li was supported by the excellent doctorial dissertation cultivation grant (2018YBZZ067 and 2019YBZZ057) from Central China Normal University.
1392
ACTA MATHEMATICA SCIENTIA
Vol.40 Ser.B
particular, the case in which the coupling is nonlinear and critical has received a great deal of attention recently, and significant progress has been made in the last thirty years since the celebrated work of Brezis and Nirenberg [10]. p Denote HRN = D1,2 (RN ) × D1,2 (RN ) with the norm k(u, v)k := kuk2 + kvk2 . We call a solution (u, v) positive if both u and v are positive, (u, v) nontrivial if u 6≡ 0 and v 6≡ 0, and (u, v) semi-trivial if (u, v) is of the form (u, 0) or (0, v). It is well known that solutions of problem (1.1) can be attained by finding nontrivial critical points of the functional Iε (u, v) = I0 (u, v) − εG (u, v) , where I0 (u, v) = and
1 2
Z
RN
Z 1 2∗ 2∗ α β 2 2 |∇u| + |∇v| − ∗ (u+ ) + (v + ) + (u+ ) (v + ) 2 RN
G(u, v) =
Z
2∗ 2∗ K(x)(u+ ) + Q(x)(v + ) RN Z Z 1 1 p+1 q+1 h(x)(u+ ) + l(x)(v + ) . + p + 1 RN q + 1 RN
1 2∗
Here u± := max{±u, 0}. We will mention some results related to problem (1.1). As we know (e.g. [11–13]), the radial function − N2−2 N −2 z0 (x) = (N (N − 2)) 4 1 + |x|2 (1.2) solves the problem
−∆u = |u|
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