Note on the Chen-Lin result with Li-Zhang method

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Note on the Chen-Lin result with Li-Zhang method

Samy Skander Bahoura

To cite this version:

Samy Skander Bahoura. Note on the Chen-Lin result with Li-Zhang method. 2010. �hal-00465213�

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NOTE ON THE CHEN-LIN RESULT WITH LI-ZHANG METHOD.

SAMY SKANDER BAHOURA

Abstract: We give a new proof of Chen-Lin result with Li-Zhang method.

1. INTRODUCTION AND RESULTS.

We set∆ =−∂11−∂22the geometric Laplacian onR².

On an open setΩofR², with a smooth boundary, we consider the following problem:

(P)

(∆u=V e^u in Ω.

0< a≤V ≤b <+∞

The previous equation is called, the Prescribed Scalar Curvature, in relation with conformal change of metrics. The functionV is the prescribed curvature.

Here, we try to find some a priori estimates for sequences of the previous problem.

Equations of the previous type were studied by many authors. We can see in [B-M], different results for the solutions of those type of equations with or without boundaries conditions and, with minimal conditions onV, for example we supposeV ≥0andV ∈L^p(Ω)orV e^u∈L^p(Ω) withp∈[1,+∞].

We can see in [B-M] the following important Theorem,

Theorem A(Brezis-Merle).If(ui)iand(Vi)i are two sequences of functions relatively to the problem(P)with,0< a≤Vi≤b <+∞, then, for all compact setKofΩ,

sup

K

ui≤c=c(a, b, m, K,Ω) if inf

Ω ui≥m.

A simple consequence of this theorem is that, if we assumeui= 0on∂Ωthen, the sequence (ui)iis locally uniformly bounded.

If, we assumeV with more regularity, we can have another type of estimates, sup + inf. It was proved, by Shafrir, see [S], that, if(ui)i,(Vi)iare two sequences of functions solutions of the previous equation without assumption on the boundary and,0 < a≤Vi ≤b <+∞, then we have the following interior estimate:

C(a/b) sup

K

ui+ inf

Ω ui≤c=c(a, b, K,Ω).

We can see in [C-L], an explicit value ofC(a/b) = ra

b.

Now, if we suppose(Vi)iuniformly Lipschitzian withAthe Lipschitz constant, then,C(a/b) = 1andc=c(a, b, A, K,Ω), see Br´ezis-Li-Shafrir [B-L-S]. This result was extended for H¨olderian sequences(Vi)i by Chen-Lin, see [C-L]. Also, we can see in [L], an extension of the Brezis- Li-Shafrir to compact Riemann surface without boundary. We can see in [L-S] explicit form, (8πm, m∈N^∗exactly), for the numbers in front of the Dirac masses, when the solutions blow- up.

On open setΩofR²we consider the following equation:

∆ui=Vie^uⁱ on Ω.

0< a≤Vi≤b <+∞, |Vi(x)−Vi(y)| ≤A|x−y|^s, 0< s <1, x, y∈Ω.

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Theorem B((Chen-Lin). For all compactK ⊂ Ωand alls ∈]0,1[there is a constantc = c(a, b, A, s, K,Ω)such that,

sup

K

ui+ inf

Ω ui≤c, ∀i.

Here we try to prove the previous theorem by the moving-plane method and Li-Zhang method.

We argue by contradiction, and we want to proof that:

∃R >0, such that 4 logR+ sup

BR(0)

u+ inf

B2R(0)u≤c=c(a, b, A), Thus, by contardition we can assume:

∃(Ri)i, (ui)i Ri→0, 4 logRi+ sup

B_Ri(0)

ui+ inf

B2Ri(0)ui→+∞, The blow-up analysis

Letx0 ∈ Ω, we want to prove the theorem locally aroundx0, we use the previous assertion withx0 = 0. The classical blow-up analysis gives the existence of the sequence(xi)i and a sequence of functions(vi)isuch that:

We set,

sup

B_Ri(0)

ui=ui(¯xi),

si(x) = 2 log(Ri− |x−¯xi|) +ui(x), andsi(xi) = sup

B_Ri(¯_xi)

si, li=1

2(Ri− |xi−x¯i|).

Also, we set:

vi(x) =ui[xi+xe^−uⁱ^(xⁱ^)/2]−ui(xi), V¯i(x) =Vi[xi+xe^−uⁱ^(xⁱ^)/2], Then,

∆vi= ¯Vie^vⁱ, vi≤2 log 2, vi(0) = 0,

vi→v= log 1

(1 + [V(0)/8]|x|²)² converge uniformly on each compact set of R² withV(0) = limi→+∞Vi(xi).

The classical elliptic estimates and the classical Harnack inequality, we can prove the previous uniform convergence on each compact ofR².

The Kelvin transform and the moving-plane method: Li-Zhang method.

For0< λ < λ1, we define:

Σλ=B(0, liMi)−B(0, λ).

First, we set :

¯

v_i^λ=v^λ_i −4 log|x|+ 4 logλ=vi

λ²x

|x|²

+ 4 log λ

|x|, V¯_i^λ= ¯Vi

λ²x

|x|²

, Mi=e^uⁱ^(xⁱ^)/2,

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and,

wλ= ¯vi−v¯^λ_i. Then,

∆¯v^λ_i = ¯V_i^λe^¯^v^λⁱ,

|y|=RminiMi

¯

v^λ_i =ui(xi+rθ)−ui(xi) + 2ui(xi)≥inf

Ω ui+ui(xi)→+∞, and,

∆(vi−v^λ_i) = ¯Vi(e^vⁱ−e^v^λⁱ) + ( ¯Vi−V¯_i^λ)e^vⁱ^λ, We have the following estimate:

|V¯i−V¯_i^λ| ≤AM_i^−s|x|^s|1− λ²

|x|²|^s, We take an auxiliary functionhλ:

Because,vi(x^λ)≤C(λ1)<+∞, we have,

hλ=C1M_i^−sλ²(log(λ/|x|)) +C2M_i^−sλ^2+s[1−( λ

|x|)^2−s], |x|> λ, withC1, C2=C1, C2(s, λ1)>0,

hλ=M_i^−sλ²(1−λ/|x|)(C1

log(λ/|x|) 1−λ/|x| +C₂^′), with,C₂^′ =C₂^′(s, λ1)>0. We can chooseC1big enough to havehλ<0.

Lemma 1: There is anλk,0>0small enough, such that, for0< λ < λk,0, we have:

wλ+hλ>0.

We have,

f(r, θ) =vi(rθ) + 2 logr, then,

∂f

∂r(r, θ) =<∇vi(rθ)|θ >+2 r, According to the blow-up analysis,

∃ r0>0, C >0, |∇vi(rθ)|θ >| ≤C, for 0≤r < r0, Then,

∃r0>0, C^′>0, ∂f

∂r(r, θ)> C^′

r , 0< r < r0, if0< λ <|y|< r0,

wλ(y) +hλ(y) =vi(y)−v^λ_i(y) +hλ(y)> C(log|y| −log|y^λ|) +hλ(y), by the definition ofhλ, we have, forC, C0>0and0< λ≤ |y|< r0,

wλ(y) +hλ(y)>(|y| −λ)[Clog|y| −log|y^λ|

|y| −λ −λ^1+sC0M_i^s], but,

|y| − |y^λ|>|y| −λ >0, and|y^λ|= λ²

|y|, thus,

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In the first step of the lemma 2, we have,

vi≥minvi=C_i¹, v_i^λ(y)≤C1(λ1, r0), if r0≤ |y| ≤RiMi, Thus, inr0≤ |y| ≤RiMiandλ≤λ1, we have,

wλ+hλ≥Ci−4 logλ+ 4 logr0−C^′λ12+s

then, ifλ→0,−logλ→+∞, and,

wλ+hλ>0, if λ < λⁱ₀, λⁱ₀ (small), andr0<|y| ≤RiMi, By the maximum princple and the Hopf boundary lemma, we have:

Lemma 2: Letλ˜kbe a positive number such that:

λ˜k = sup{λ < λ1, wλ+hλ>0 in Σλ.}.

Then,

λ˜k=λ1.

The blow-up analysis gives the follwing inequality for the boundary condition, Fory=|y|θ=RiMiθ, we have,

wλⁱ(|y|=RiMi) +hλⁱ(|y|=RiMi) =

=ui(xi+Riθ)−ui(xi)−vi(y^λⁱ,|y|=RiMi)−4 logλ+4 log(RiMi)+C(s, λ1)M_i^−sλ^2+s[1−( λ RiMi

)^2−s], 4 log ¯Ri+ui(xi) + inf

Ω ui →+∞, which we can write,

wλⁱ(|y|=RiMi) +hλⁱ(|y|=RiMi)≥min

Ω ui+ui(xi) + 4 logRi−C(s, λ1)→+∞, because,0< λ≤λ1,

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References.

[B-G] L. Boccardo, T. Gallouet. Nonlinear elliptic and parabolic equations involving measure data. J. Funct. Anal. 87 (1), (1989), 149-169.

[B-L-S] H. Brezis, YY. Li and I. Shafrir. A sup+inf inequality for some nonlinear elliptic equations involving exponential nonlinearities. J.Funct.Anal.115 (1993) 344-358.

[B-M] H. Brezis, F. Merle. Uniform estimates and Blow-up Behavior for Solutions of−∆u= V(x)e^u in two dimension. Commun. in Partial Differential Equations, 16 (8 and 9), 1223- 1253(1991).

[C-Li] W. Chen, C. Li. A priori Estimates for solutions to Nonlinear Elliptic Equations. Arch.

Rational. Mech. Anal. 122 (1993) 145-157.

[C-L] C-C. Chen, C-S. Lin. A sharp sup+inf inequality for a nonlinear elliptic equation inR². Commun. Anal. Geom. 6, No.1, 1-19 (1998).

[L] YY. Li, Harnack type Inequality, the Methode of Moving Planes. Commun. Math. Phys.

200 421-444.(1999).

[L-S] YY. Li, I. Shafrir. Blow-up Analysis for Solutions of−∆u=V e^uin Dimension Two.

Indiana. Math. J. Vol 3, no 4. (1994). 1255-1270.

[M-W] L. Ma, J-C. Wei. Convergence for a Liouville equation. Comment. Math. Helv. 76 (2001) 506-514.

[R] W. Rudin. Real and Complex Analysis.

[S] I. Shafrir. A sup+inf inequality for the equation−∆u=V e^u. C. R. Acad.Sci. Paris S´er. I Math. 315 (1992), no. 2, 159-164.

E-mail address:samybahoura@yahoo.fr

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