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Sharp quantization for Lane-Emden problems in
dimension two
Pierre-Damien Thizy
To cite this version:
DIMENSION TWO
PIERRE-DAMIEN THIZY
Abstract. In this short note, we prove a sharp quantization for positive solutions of Lane-Emden problems in a bounded planar domain. This result has been conjectured by De Marchis, Ianni and Pacella [6, Remark 1.2].
Introduction
Let Ω be an open, non-empty, connected and bounded subset of R2with smooth
boundary ∂Ω and let ∆ = −(∂xx+ ∂yy) be the (positive) laplacian. In this paper,
we are interested in the asymptotic behavior as p → +∞ of a sequence (up)p of
smooth functions, positive in Ω, and satisfying the so-called Lane-Emden problem (
∆up= |up|p−1up in Ω ,
up= 0 on ∂Ω ,
(0.1) together with the bounded-energy type assumption
p Z
Ω|∇u
p|2ds = O(1) , (0.2)
for all p. Up to now, the most general results on this problem were obtained by De Marchis, Ianni and Pacella [6]. In particular, for such a given (up)p satisfying
(0.1)-(0.2), it is proved in [6] that, up to a subsequence, there exists an integer n ≥ 1 and a subset B = {x1, ..., xn} of Ω such that the following quantization
lim p→+∞p Z Ω|∇u p|2ds = 8π n X j=1 m2j (0.3)
holds true, where the mj’s can be obtained through
mj = lim
β→0+p→+∞lim kupkC0(Bxj(β)), (0.4)
for j ∈ {1, ..., n}, where Bxj(β) is the ball of center xj and radius β. Observe that,
in particular, the xj’s are not in ∂Ω. It is also proved in [6] that we necessarily
have that
mj ≥√e , (0.5)
for all j ∈ {1, ..., n} and that lim
p→+∞up= 0 in C 2
loc( ¯Ω\B) . (0.6)
In [6, Remark 1.2], it is conjectured that we must have equality in (0.5), so that, in some sense, the constant 8πe plays here the same role as the Sobolev constant
Date: February 2018.
1991 Mathematics Subject Classification. 35B05, 35B06, 35J91.
2 PIERRE-DAMIEN THIZY
in dimensions greater than 2 (see Struwe [18]). This is the point in the following theorem.
Theorem 0.1. Let Ω be a smooth bounded domain of R2. Let(u
p)p be a sequence
of smooth functions positive in Ω, and satisfying (0.1) and (0.2). Then, up to a subsequence, there exists an integern ≥ 1 such that
lim
p→+∞p
Z
Ω|∇u
p|2ds = (8πe) × n . (0.7)
Moreover, there exists a subset {x1, ..., xn} of Ω such that
mj =√e , (0.8)
wheremj is given by (0.4), for all j ∈ {1, ..., n}.
In addition, by [6], we get from (0.8) that lim p→+∞pup= 8π √ e n X j=1 Gxj in C 2 loc( ¯Ω\B) , (0.9) and that ∇xj Hxj(xj) + X i6=j Gxi(xj) = 0 , (0.10) for all j ∈ {1, ..., n}, where G is the Green’s function of ∆ with Dirichlet boundary conditions and where H is its regular part, which is smooth in Ω2 and given by
Gx(y) = 1 2πlog 1 |x − y|+ Hx(y) , for all x 6= y.
Concerning the previous works, Ren and Wei [16] and [17] where able to prove that (0.7) with n = 1 holds true if the up’s are minimizers, i.e. if we assume in
addition that up is proportional to a solution of the problem
min {v∈H1 0 s.t. R Ωvpds=1} Z Ω|∇v| 2ds.
Answering to a former question, Adimurthi and Grossi [1] were able to prove that lim
p→+∞kupkC0(Ω)=
√
e , (0.11)
in the case of minimizers, while they discovered the way to perform the first rescal-ing for the up’s as p → +∞, and the key link with the Liouville equation. Observe
that (0.4), (0.6) and (0.8) clearly imply (0.11) in general case. Now in the ra-dial case where Ω is a disk, observe that the up’s are necessarily minimizers, since
(0.1) admits only one solution (see Gidas-Ni-Niremberg [11] and the nice survey by Pacella [15]); according to the previous discussion, we necessarily then have that n = 1 in (0.7). In contrast, if Ω is not simply connected, Esposito, Musso and Pis-toia [9] were able to prove that, for all given integer n ≥ 1, there exists a sequence of positive functions (up)psatisfying (0.1)-(0.2) such that (0.7) holds true, together
with (0.8)-(0.11). Thus, in some sense, Theorem 0.1 is sharp. We mention that very interesting complementary results were obtained recently by Kamburov and Sirakov [14]. At last, we also mention that, even if the situation is far from being as well understood in the nodal case, where we no longer assume that the up’s are
results were obtained.
To conclude, as explained in De Marchis, Ianni and Pacella [4], the techniques to get the quantization result in [6] are not without similarity with the ones devel-oped by Druet [7] to get the analogue quantization for 2D Moser-Trudinger critical problems. Both results [6, 7] can be improved by showing that all the blow-up points necessarily carry the minimal energy. It is done here in the Lane-Emden case and in Druet and Thizy [8] in the Moser-Trudinger case. Unfortunately, the authors of [8] were not able to find an as easy argument as here, in the more tricky Moser-Trudinger critical case.
1. Proof of Theorem 0.1
Let (up)p be a sequence of smooth functions, positive in Ω and satisfying
(0.1)-(0.2). Then by [6], (0.3)-(0.6) hold true. Thus, the proof of (0.7)-(0.8), i.e. that of Theorem 0.1, reduces to the proof of
lim
p→+∞kupkC0(Ω)≤
√
e . (1.1)
Here and in the sequel, we argue up to a subsequence. Now, let (yp)p be a sequence
in Ω such that up(yp) = kupkC0(Ω), for all p. By (0.4)-(0.6), we have that
lim
p→+∞d(yp, ∂Ω) := 2δ0> 0 , (1.2)
where d(y, ∂Ω) denotes the distance from y to ∂Ω. Now, let µp> 0 be given by
µ2p p up(yp)p−1 = 8 . (1.3)
By (0.4) and (0.5), we get from (1.3) that log 1 µ2 p = p log up(yp) 1 + O log p p , (1.4)
and in particular, that µp→ 0 as p → +∞. Let τp be given by
up(yp+ µpy) = up(yp) 1 − 2τpp(y) , so that τp≥ 0 and τp(0) = 0 , (1.5)
by definition of (yp)p. By (0.1) and (1.3), we have that
∆(−τp) = 4 1 −2τpp p in Ωp:=Ω − yp µp , (1.6)
so that, by (1.5), positivity of up and concavity of the log function, we get that
0 < ∆(−τp) ≤ 4 . (1.7)
By (1.2), (1.5)-(1.7) and standard elliptic theory, including the Harnack principle, we get that there exists a function τ∞∈ C2(R2) such that
lim
p→+∞τp= τ∞ in C 2
loc(R2) , (1.8)
and then that
4 PIERRE-DAMIEN THIZY
using also that ∇τp(0) = 0, by definition of (yp)p. Let R > 0 be given. Integrating
by parts, using (1.3), (1.8), (0.2), ∆up≥ 0 and up≥ 0, we get that
lim inf
p→+∞(8up(yp) 2)Z
B0(R)
4 exp(−2τ∞)dy ≤ lim p→+∞p Z Byp(Rµp) (∆up)updy , ≤ lim p→+∞p Z Ω|∇u p|2dy < +∞ . (1.10)
By using that up(yp)2 ≥ (1 + o(1))e , and by observing that the above RHS does
not depend on R > 0, which can be arbitrarily large, we get that Z
R2exp(−2τ
∞)dy < +∞ . (1.11)
By Chen and Li [2], (1.9) and (1.11) imply that
τ∞= log(1 + | · |2) . (1.12)
Then, we let tp be given by
tp(y) = log 1 +|y − yp| 2 µ2 p .
From now on, if f is a given continuous function in Ω, we let ¯f be the unique continuous function in [0, d(yp, ∂Ω)) given by
¯
f (r) = 1 2πr
Z
∂Byp(r)
f dσ , for all r ∈ (0, , d(yp, ∂Ω)) .
Let (rp)p be any sequence such that rp ∈ [0, d(yp, ∂Ω)) for all p. By (0.1), (1.3),
(1.8), (1.12) and Fatou’s lemma, we get that − 2πrp d¯up dr (rp) = Z Byp(rp) (∆up) 2πrdr , ≥ 2upp(yp) Z rp/µp 0 8πr dr (1 + r2)2 + o (rp/µp)2 1 + (rp/µp)2 ! , (1.13)
using that the laplacian commutes with the average in spheres. Then, using the fundamental theorem of calculus and ¯up(0) = up(yp), we easily get from (1.13) that
¯ up(rp) ≤ up(yp) 1 −2¯tpp(rp)+ o ¯ tp(rp) p . (1.14) Picking now rp= δ0for all p, according to (1.2), we get from (1.14) that
¯ up(δ0) ≤ up(yp) 1 − 2 log up(yp) (1 + o(1)) + O 1 p , (1.15)
by (1.4), writing merely log 1 µ2
p + log µ
2
p+ δ02 = ¯tp(δ0). Since ¯up ≥ 0 and since
up(yp) = kupkC0(Ω), we easily get from (1.15) that (1.1) holds true. As explained
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