Anisotropic Moser-Trudinger inequality involving Ln norm

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Anisotropic Moser-Trudinger inequality involving Ln norm Changliang Zhou 1 School of Science, East China University of Technology, Nanchang, China Received 17 December 2018; revised 27 August 2019; accepted 15 November 2019

Abstract The paper is concerned about a sharp form of Anisotropic Moser-Trudinger inequality which involves Ln norm. Let λ1 () =

inf

u∈W01,n (),u≡0

||F (∇u)||nLn () /||u||nLn ()

be the first eigenvalue associated with n-Finsler-Laplacian. Using blow-up analysis, we obtain that  sup

1

e

n

λn (1+α||u||nLn () ) n−1 |u| n−1

dx

u∈W01,n (),||F (∇u)||Ln () =1  n

1

is finite for any 0 ≤ α < λ1 (), and the supremum is infinite for any α ≥ λ1 (), where λn = n n−1 κnn−1 (κn is the volume of the unit wulff ball) and the function F is positive, convex and homogeneous of degree 1, and its polar F o represents a Finsler metric on Rn . Furthermore, the supremum is attained for any 0 ≤ α < λ1 (). © 2019 Published by Elsevier Inc. MSC: 35A05; 35J65

E-mail address: [email protected]. 1 The author is supported partially by the funding for the Doctoral Research of ECUT under grant NO. DHBK2018053

and NSFC of China (No. 11771285). https://doi.org/10.1016/j.jde.2019.11.066 0022-0396/© 2019 Published by Elsevier Inc.

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Keywords: Moser-Trudinger inequality; n-Finsler-Laplacian; Blow-up analysis

1. Introduction Let  ⊂ Rn be a smooth bounded domain. The Sobolev embedding theorem states that is embedded in Lp () for any p > 1, or equivalently, using the Dirichlet norm  1 u W 1,n () = (  |∇u|n dx) n on W01,n (), W01,n () 0

 |u|p dx < +∞.

sup u∈W01,n (),||∇u||Ln () ≤1 

But it is well known that W01,n () is not embedded in L∞ (). Hence, one is led to look for a function g(s) : R → R+ with maximal growth such that  sup g(u)dx < +∞. u∈W01,n (),||∇u||Ln () ≤1



The Moser-Trudinger inequality states that the maximal growth is of exponential type, which was shown by Pohozhaev [26], Trudinger [27] and Moser [20]. This inequality says that  n n−1 sup eα|u| dx < +∞ (1) u∈W01,n (),||∇u||Ln () ≤1



1

n−1 for any α ≤ αn , where αn = nωn−1 and ωn−1 is the surface area of the unit ball in Rn . The inequality is optimal, i.e. for any α > αn there exists a sequence of {u } in W01,n () with ||∇u ||Ln () ≤ 1 such that



eα|u |

n n−1

dx → ∞

as  → 0.



On the other hand, for any fixed u ∈ W01,n (), it is also known that  eα|u|

n n−1

dx < +∞



for any α > 0. Another interesting question about Moser-Trudinger inequalities is whether extremal functions exist or not. The first result in this direction is due to Carleson and Chang [6], who proved that the supremum is attained when  is a unit ball in Rn . Then Struwe [22] got the existence of extremals for  close to a ball. Struwe’s technique was then used and extended by Flucher [9] to  which is the more general bounded smooth domain in R2 . Later, Lin [17] generalized the existence result to a bounded smooth domain in dimension-n. Recently Mancini and Martinazzi

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[21] reproved the Carleson and Chang’s result by using a new method based on the Dirichlet energy, also allowing for perturbations of the functional. Even Thizy [28] also gave examples in which slightly perturbed Moser-Trudinger inequalities do not admit extremals (as conjectured in [21]), hence shown that the existence of extremal for the Moser-Trudinger inequality should not be taken for granted and holds only under some rigid conditions. Actually, the inequality (1) is viewed as a n-dimensional analog of the Sobolev inequality, and it plays an important role in analytic problems and in geometric problems. Now there are many generalizations of the classical Moser-Trudinger inequality (1), see for instance [1,7,14,16,19,33,35–41] and the references therein. In 2012, Wang and Xia [31] proved the following Moser-Trudinger type inequality  eλ|u|

n n−1

dx ≤ C(n)||

(2)



for all u ∈ W01,n () and



1

n

≤ 1. Here λ ≤ λn = n n−1 κnn−1 , λn is optimal in the sense n  n−1 that if λ > λn we can find a sequence {uk } such that  eλ|uk | dx diverges. Later, in [42] and [43] there have shown that the supremum is attained when  is bounded domain in Rn . Adimurthi and Druet [1], Y.Y. Yang [35] have proved that, when λ1 () > 0 be the first eigenvalue of the n-Laplacian with Dirichlet boundary condition in , then (1) For any 0 ≤ α < λ1 (), F

n (∇u)dx

 sup u∈W01,n (),||∇u||nLn () =1

1

n

e

αn |u| n−1 (1+α||u||nLn () ) n−1

e

αn |u| n−1 (1+α||u||nLn () ) n−1

dx < +∞.



(2) For any α ≥ λ1 (),  sup u∈W01,n (),||∇u||nLn () =1

1

n

dx = +∞.



Furthermore, when 0 ≤ α < λ1 (), the supremum is also attained. These results were extended by Yang [37] to compact Riemann surface case and by Lu-Yang [19] and Zhu [41] to a version involving Lp norm for any p > 1. For simplicity, we introduce the notations  Jλα (u) =

1

e

n

λ(1+α||u||nLn () ) n−1 |u| n−1

dx, H = {u ∈ W01,n () : ||F (∇u)||Ln () = 1}.

(3)



Let λ1 () be the first eigenvalue of Finsler-n-Laplacian with Dirichlet boundary condition in . It is defined by λ1 () =

inf

u∈W01,n (),u=0

||F (∇u)||nLn () ||u||nLn ()

,

(4)

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from [4], we can know that λ1 () > 0 and it is also achieved uniquely by a positive function ϕ satisfying 

−Qn ϕ = λ1 ()|ϕ|n−2 ϕ, ϕ=0

in  on ∂,

where −Qn is Finsler-Laplacian operator that can be founded in section 2. In this paper we state the following: Theorem 1.1. Let  ⊂ Rn be a smooth and bounded domain and let λ1 () > 0 be the first eigenvalue of the Finsler-n-Laplacian with Dirichlet boundary condition in . Then we have (1) For any 0 ≤ α < λ1 (), sup Jλαn (u) < +∞. (2) For any α ≥ λ1 (),

u∈H sup Jλαn (u) = +∞. u∈H

Theorem 1.2. Let  ⊂ Rn be a smooth and bounded domain. When n ≥ 3, for any 0 ≤ α < λ1 (), supu∈H Jλαn (u) is attained by some C 1 maximizer. In other words, there exists uα ∈ H ∩ C 1 () such that Jλαn (uα ) = supu∈H Jλαn (u). When F (x) = |x|, Adimurthi and Druet [1], Y.Y. Yang [35] have proved the above theorem. But the F (x) = |x|, it is more different, need much more delicate work. Now we describe the main idea to prove Theorem 1.1 and Theorem 1.2. The proof of point (2) of Theorem 1.1 is base on test functions computations which are present in Section 2. The point of point (1) of Theorem 1.1 is based on the blow up analysis. The proof of Theorem 1.2 is based on two facts: an upper bound of Jλαn on H can be derived under the assumption that blowing up occur; a sequence of functions φ ∈ H can be constructed to show that the above upper bound is not an upper bound. This contradiction implies that no blowing up occur, and then Theorem 1.2 holds. Though the method we carry out blowing up analysis is routine, we will encounter new difficulties when 0 < α < λ1 (). We organize this paper as follows. In Section 2, we give some notes about anisotropic function F (x) and the properties of the function F (x), moreover, we prove point (2) of Theorem 1.1. We use blowing up analysis to prove point of (1) of Theorem 1.1 in section 3 and section 4. An upper bound of Jλαn is derived, moreover a sequence of functions is constructed to reach a contradiction in section 5, which completes the proof of Theorem 1.2. In section 6, we show the asymptotic representation of certain Green function which has been used in Section 5. 2. Notations and preliminaries In this section, we will give the notations and preliminaries. Throughout this paper, let F : Rn → R be a nonnegative convex function of class C 2 (Rn \{0}) which is even and positively homogenous of degree 1, so that F (tξ ) = |t|F (ξ )

for any

t ∈ R,

ξ ∈ Rn .

1  A typical example is F (ξ ) = ( i |ξi |q ) q for q ∈ [1, ∞). We further assume that

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F (ξ ) > 0

for any

5

ξ = 0.

Thanks to homogeneity of F , there exist two constants 0 < a ≤ b < ∞ such that a|ξ | ≤ F (ξ ) ≤ b|ξ |. Usually, we shall assume that the H ess(F 2 ) is positive definite in Rn \{0}. Then by Xie and Gong [34], H ess(F n ) is also positive definite in Rn \{0}. If we consider the minimization problem  F n (∇u)dx,

min

u∈W01,n ()



whose Euler-Lagrange equation contains an operator of the form Qn u :=

n  ∂ (F n−1 (∇u)Fξi (∇u)). ∂xi i=1

Note that these operators are not linear unless F is the Euclidean norm in dimension two. We call this nonlinear operator as n-Finsler-Laplacian. This operator Qn was studied by many mathematicians, see [2,4,10,11,31,34] and the references therein. Consider the map φ : S n−1 → Rn ,

φ(ξ ) = Fξ (ξ ).

Its image φ(S n−1 ) is a smooth, convex hypersurface in Rn , which is called Wulff shape of F . Let F o be the support function of K := {x ∈ Rn : F (x) ≤ 1}, which is defined by F o (x) := sup x, ξ . ξ ∈K

It is easy to verify that F o : Rn → [0, +∞) is also a convex, homogeneous function of class of C 2 (Rn \{0}). Actually F o is dual to F in the sense that x, ξ  , ξ =0 F (ξ )

F o (x) = sup

x, ξ  . o ξ =0 F (ξ )

F (x) = sup

One can see easily that φ(S n−1 ) = {x ∈ Rn |F o (x) = 1}. We denote WF := {x ∈ Rn |F o (x) ≤ 1} and κn := |WF |, the Lebesgue measure of WF . We also use the notion Wr (0) := {x ∈ Rn |F o (x) ≤ r}. We call Wr (0) a Wulff ball of radius r with center at 0. For later use, we give some simple properties of the function F , which follows directly from the assumption on F , also see [5,11,32]. Lemma 2.1. We have (i) |F (x) − F (y)| ≤ F (x + y) ≤ F (x) + F (y); (ii) C1 ≤ |∇F (x)| ≤ C, and C1 ≤ |∇F o (x)| ≤ C for some C > 0 and any x = 0;

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x, ∇F (x) = F (x), x, ∇F o (x) = F o (x) for any x = 0; F (∇F o (x)) = 1, F o (∇F (x)) = 1 for any x = 0; F o (x)Fξ (∇F o (x)) = x for any x = 0; Fξ (tξ ) = sgn(t)Fξ (ξ ) for any ξ = 0 and t = 0.

Next we describe the isoperimetric inequality and co-area formula with respect to F . For a domain  ⊂ Rn , a subset E ⊂  and a function of bounded variation u ∈ BV (), we define the anisotropic bounded variation of u with respect to F is 

 |∇u|F = sup{



u divσ dx, σ ∈ C01 (; Rn ), F o (σ ) ≤ 1}.



We set anisotropic perimeter of E with respect to F is  PF (E) :=

|∇XE |F dx, 

where XE is the characteristic function of the set E. It is well known (also see [10]) that the co-area formula 

∞ |∇u|F =



(5)

PF (|u| > t)dt 0

and the isoperimetric inequality 1

1

PF (E) ≥ nκnn |E|1− n

(6)

hold. Moreover, the equality in (6) holds if and only if E is a Wulff ball. In the sequel, we will use the convex symmetrization with respect to F . The convex symmetrization generalizes the Schwarz symmetrization (see [30]). It was defined in [2] and will be an essential tool for establishing the Lions type concentration-compactness under the anisotropic Dirichlet norm. Let us consider a measured function u on  ⊂ Rn . The one dimensional decreasing rearrangement of u is u∗ = sup{s ≥ 0 : |{x ∈  : |u(x)| > s}| > t},

for t ∈ R.

The convex symmetrization of u with respect to F is defined as u (x) = u∗ (κn F o (x)n ),

for x ∈ ∗ .

Here κn F o (x)n is just Lebesgue measure of a homothetic Wulff ball with radius F o (x) and ∗ is the homothetic Wulff ball centered at the origin having the same measure as . Throughout this paper, we assume that  is bounded smooth domain in Rn with n ≥ 2. The following properties about Finsler-Laplacian can be found in [43]:

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Lemma 2.2. Assume that u ∈ W01,n () is a solution to the equation −Qn (u) = f.

(7) 1

If f ∈ Lq () for some q > 1, then ||u||L∞ () ≤ C||f ||Ln−1 q () , where C is only depends on a, b, n, , q. Set  P=

(1 −



F



1 n (∇u)dx)− n−1



∞

F n (∇u)dx < 1,

F

n (∇u)dx

= 1.

Lemma Let u ∈ W01,n (), u = 0. Assume that {uk } is a sequence of functions W01,n () such  2.3. n that  F (∇uk )dx ≤ 1, and uk  u weakly in W01,n (),

uk → u

a.e. in .

Then for every p < P , there exists a constant C = C(n, p) such that for each k 

n

1

n

exp(n n−1 κnn−1 p|uk | n−1 )dx ≤ C.

(8)



Moreover, this conclusion fails if p ≥ P . Test functions computations In this subsection we will prove the conclusion (2) of Theorem 1.1. We will build explicit test functions to show the unboundedness of Moser-Trudinger function under large parameter. Since the Moser-Trudinger inequality is invariant under translation, we may assume that 0 ∈  and W1 ⊂ . We now fix some xδ ∈ W1 such that F o (xδ ) = δ. Choosing t such that tn log 1 → ∞ and tn+1 log 1 → 0. Set

ϕ (x) =

⎧ n ⎪ ( n log 1 ) n−1 , ⎪ ⎨ λn n ⎪ ⎪ ⎩

( λnn

log

F o (x) ≤ ,

1 n−1 (log δ−log F o (x))−t ϕ(xδ )(log −log F o (x)) )

log δ−log 

t [ϕ(xδ ) + θ (x)(ϕ(x) − ϕ(xδ )),

,

 < F o (x) ≤ δ, F o (x) > δ.

In above definition of ϕ (x), ϕ is the eigenvalue function. θ (x) ∈ C 2 () is a cut-off function satisfy |∇θ (x)| ≤ Cδ and ⎧ ⎨ 0, θ (x) = θ ∈ (0, 1), ⎩ 1, Let δ =

1 1 tn log 1 

F o (x) ≤ δ δ < F o (x) < 2δ F o (x) ≥ 2δ.

, it is easy to see  < δ if  is small enough. We obtain that

(9)

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n

| − ( λnn log 1 ) n−1 + t ϕ(xδ )|n

F (∇ϕ (x))dx = n


|F o (x)|n (log δ − log )n


dx

n

=

nκn | − ( λnn log 1 ) n−1 + t ϕ(xδ )|n

(log δ − log )n−1 n+1 1 1 n−1 = 1 − n n κnn (log )− n t ϕ(xδ )(1 + o (1)), 

where o (1) → 0 as  → 0. We also have   F n (∇ϕ (x))dx = tn δ≤F o (x)≤2δ

|θ (x)∇ϕ + ∇θ (x)(ϕ(x) + ϕ(xδ ))|n dx

δ≤F o (x)≤2δ

= tn O(δ n ) and



 n

F (∇ϕ (x))dx

= tn

F o (x)≥2δ

F n (∇ϕ(x))dx

F o (x)≥2δ

= tn (1 + O(δ n )). Summing the above integral estimates for F n (∇ϕ ) up, we have  n+1 1 1 n−1 F n (∇ϕ (x))dx = 1 − n n κnn (log )− n t ϕ(xδ )(1 + o (1)) + tn (1 + O(δ n )).  

Then −

n

||F (∇ϕ (x))||Lnn−1 () = 1 + Set v (x) =

ϕ (x) ||F (∇ϕ (x))||Ln () ,

n+1

1 n n 1 n−1 1 n κnn (log )− n t ϕ(xδ )(1 + o (1)) − t (1 + O(δ n )). n−1  n−1 

then ||F (∇v (x))||Ln () = 1. Furthermore,

λ1 ()||v (x)||nLn () λ1 ()tn ≥ ||F (∇ϕ (x))||nLn ()

 |ϕ(x)|n dx F o (x)≥2δ

≥ λ1 ()tn [||ϕ(x)||nLn () + O(δ n )][1 + n

n+1 n

1 1 n−1 κnn (log )− n t ϕ(xδ )(1 + o (1)) 

− tn (1 + O(δ n ))] = tn (λ1 ()||ϕ(x)||nLn () + O(δ n ))(1 + O(tn )) = tn (1 + O(tn ) + O(δ n )), where we have used λ1 ()||ϕ(x)||nLn () = 1.

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Next we establish the integral estimates on the domain of {x ∈  : F o (x) < }. We have −n 1 n 1 1 λn (1 + λ1 ()||v ||nLn () ) n−1 |v | n−1 ≥ n log (1 + λ1 ()||v ||nLn () ) n−1 ||F (∇ϕ )||Ln−1 n ()  1 1 = n log (1 + tn (1 + O(tn ) + O(δ n ))) n−1  n+1

1 n n 1 n−1 ·(1 + κnn (log )− n t ϕ(xδ )(1 + o (1)) n−1  1 n t (1 + O(δ n ))) − n−1  2n+1

1 1 1 1 n n κnn (log ) n t ϕ(xδ )(1 + o (1)) = n log +  n−1  n 1 1 n log tn (1 + O(δ n )) + log tn (1 + O(tn )) − n−1  n−1  n +O(δ ) + o (1) 2n+1

1 1 1 1 n n κnn (log ) n t ϕ(0)(1 + o (1)), = n log +  n−1 

where the fact that ϕ(xδ ) = ϕ(0) + o (1) is applied. Note that log 1 tn O(δ n ) = o (1). Considering the above estimates, we deduce that 

2n+1

1 n exp(λn (1 + λ1 ()||v ||nLn () ) n−1 |v | n−1 )dx



1 n n 1 1 ≥ Cexp[ κnn (log ) n t ϕ(0)(1 + o (1))] n−1 

→ +∞ 1

as  → 0, since ϕ(0) > 0 and (log 1 ) n t → +∞. Here C is a positive constant independent of . The conclusion (2) in Theorem 1.1 holds. 3. Maximizers of the subcritical case In this section, we will show the existence of the maximizers for Moser-Trudinger functionals in the subcritical case. We begin with the following existence of the maximizers of the subcritical Moser-Trudinger function. Proposition 3.1. For any small  and 0 ≤ α < λ1 , there exists some u ∈ C 1 () ∩ H satisfying Jλαn − (u ) = sup Jλαn − (u). u∈H

Proof. For any fixed , let {u,j } ⊂ H be a sequence such that lim Jλαn − (u,j ) = sup Jλαn − (u).

j →+∞

u∈H

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Since u,j is bounded in W01,n (), there exists a subsequence of u,j (we do not distinguish subsequence and sequence in the paper) such that u,j  u weakly in W01,n (), u,j → u strongly in Lp (), for any p ≥ 1, u,j → u a.e.  as j → +∞. Hence 1

n

gj := exp[(λn − )(1 + α||u,j ||nLn () ) n−1 |u,j | n−1 ] → g 1

n

:= exp[(λn − )(1 + α||u ||nLn () ) n−1 |u | n−1 ] a.e. in . We claim that u ≡ 0, suppose not, 1 + α||u,j ||nLn () → 1, from which one can see that gj is bounded in Lp () for some p > 1 and gj → 1 in L1 (). Hence supu∈H Jλαn − (u) = ||, which is impossible. Therefore u ≡ 0. Thanks to Lemma 2.3, for any q < 1/(1 − 1 ||F (∇u )||nLn () ) n−1 , we have  j →+∞

n

exp[λn q|u,j | n−1 ]dx < +∞.

lim sup 

Due to (4), we get 1 + α||u ||nLn () <

1 1 − ||F (∇u )||nLn ()

for 0 ≤ α < λ1 . Thus, gj is bounded in Ls () for some s > 1. Since gj → g a.e. in , we infer that gj → g strongly in L1 () as j → +∞. Therefore, the extremal function is attained for the case of λn −  and ||∇u ||nLn () = 1. Clearly we can choose u ≥ 0. It is not difficult to check that the Euler-Lagrange equation of u is ⎧ n 1 ⎪ −1 u n−1 eα un−1 + γ un−1 , ⎪ u = β λ −Q n    ε  ⎪  ⎪ ⎪ 1,n ⎪ n () = 1, u ≥ 0, u ∈ W (), ||F (∇u )|| ⎪   L ⎪ 0 ⎪ 1 ⎨ α = (λn − )(1 + α||u ||nLn () ) n−1 , ⎪ β = (1 + α||u ||nLn () )/(1 + 2α||u ||nLn () ). ⎪ ⎪ ⎪ ⎪ ⎪ γ = α/(1 + 2α||u ||nLn () ), ⎪ ⎪ n n ⎪  n−1 ⎩ λ =  un−1 eα u dx. 1

n n−1

n−1 α u Since β λ−1  u e

(10)

and γε un−1 are bounded in Ls () for some s > 1, then by Lemma 2.2,  1

n n−1

n−1 α u + γε un−1 ∈ L∞ (). Then by Theorem 1 in we have u ∈ L∞ (). It implies β λ−1  u e  1,α [18], we easily get u ∈ C () for some α ∈ (0, 1), which implies that u ∈ C 1 (). 2

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The following observation is important. Lemma 3.2. For any α ∈ [0, λ1 ()), we have lim Jλαn − (u ) = sup Jλαn − (u).

→0

u∈H

Proof. Obviously, lim Jλαn − (u ) ≤ sup Jλαn − (u).

→0

u∈H

On the other hand, for any u ∈ W01,n () with ||F (∇u)||Ln () ≤ 1, from Fatou’s Lemma and Proposition 3.1 we have  e



1

n

λn |u| n−1 (1+α||u||nLn () ) n−1

dx ≤ lim inf →0



 ≤ lim inf

(λn −)|u| n−1 (1+α||u||nLn () ) n−1

dx

 1

n

e

→0

1

n

e

(λn −)|u | n−1 (1+α||u ||nLn () ) n−1

dx,



which implies lim Jλαn − (u ) ≥ sup Jλαn − (u).

→0

Hence the result holds.

u∈H

2

4. Blow-up analysis In this section, we consider the convergence of the maximizing sequence in section 3. There are two cases. The one case is that M = max u = u (x ) is bounded. In this case, it is clear that u is bounded in W01,n (). Then we can assume without loss of generality u  u0

weakly in W01,n (),

u → u0

strongly in Lq (), ∀q ≥ 1,

a.e. in . u → u0  Since, for any u ∈ W01,n () with  F n (∇u)dx ≤ 1, by the Lebesgue dominated convergence theorem we have 

n

e



1

λn |u| n−1 (1+α||u||nLn () ) n−1

dx = lim

→0



1

n

e

(λn −)|u| n−1 (1+α||u||nLn () ) n−1

dx



 

≤ lim

n

e

→0  

1

(λn −)|u | n−1 (1+α||u ||nLn () ) n−1

dx

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C. Zhou / J. Differential Equations ••• (••••) •••–•••

12

 =

n

e

1

λn |u0 | n−1 (1+α||u0 ||nLn () ) n−1

dx.



Hence u0 is the desired maximizer. The other case is that M = u (x ) → +∞ and x → x0 as  → 0. In this case, the maximizing sequence u blows up as  → 0, where x0 is called the blow-up point. In the sequel, we will analysis the blow-up behaviors of u . First, by an inequality et ≤ 1 + tet , we have  || <

eα |u |

n n−1

dx ≤ || + α λ .



This leads to lim inf→0 λ > 0. Case 1. x0 lies in the interior of . Lemma 4.1. There holds u0 = 0 and F n (∇u )dx  δx0 in the sense of measure as  → 0, where δx0 is the dirac measure at x0 . Proof. Suppose u0 ≡ 0, then for any α ∈ [0, λ1 ()), we have 1 + α||u ||nLn () → 1 + α||u0 ||nLn () ≤ 1 + ||F (∇u0 )||nLn () <

1 . 1 − ||F (∇u0 )||nLn ()

n n−1

Hence eα |u | is bounded in Ls () for some s > 1 provided  is sufficiently small. Lemma 2.2 implies that u is uniformly bounded in . It contradicts the assumption that M → +∞. Assume that F n (∇u )dx  μ in the sense of measure as  → 0, if μ = δx0 , we claim that there exists a cut-off function φ(x) ∈ C01 (), which is supported in Wr (x0 ) for some r > 0 with 0 < φ(x) < 1 in Wr (x0 )\W 2r (x0 ) and φ(x) = 1 in W 2r (x0 ) satisfying  φF n (∇u )dx ≤ 1 − η Wr (x0 )

for some η > 0 and small enough . We prove the claim by contradiction. Suppose by the contradiction that there exist sequences of ηi → 0 and ri → 0 as i → +∞ such that  φi F n (∇u )dx > 1 − ηi , Wri (x0 )

for every φi ∈ C01 (Wri (x0 )) and φi = 1 in W ri (x0 ). Then 2

 φi F n (∇u )dx > 1 − ηi . W ri (x0 ) 2

(11)

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[m1+; v1.304; Prn:27/11/2019; 13:54] P.13 (1-35)

C. Zhou / J. Differential Equations ••• (••••) •••–•••

13

Taking i → ∞, the left hand side of (11) converges to 0. However, 1 − ηi → 1, this contradiction leads to the claim. Since u → 0 strongly in Lq () for any q > 1, we may assume that  F n (∇(φu ))dx ≤ 1 − η Wr (x0 ) n n−1

provided  is sufficient small. By (2), eλn (φu ) is uniformly bounded in Ls (Wr0 (x0 )) for some s > 1 and 0 < r0 < r. Applying Lemma 2.2, u is uniformly bounded in W r0 (x0 ), which 2 contradicts the fact that M → +∞ again. Therefore, F n (∇u )dx  δx0 as  → 0. 2 Now we set n

rεn

n−1 − n = λε β−1 Mε n−1 e−α M .

(12)

Fixed any δ ∈ (0, λ2n ), by the expression of r in (12) and λ in (10), we have n

n − n−1

n

rn exp{δMn−1 } = λ β−1 M

n

exp{−α Mn−1 }exp{δMn−1 }  n n n n n−1 −1 − n−1 = β M exp{(δ − α )M } un−1 exp{α un−1 }dx 

≤

− n β−1 M n−1 exp{(2δ

≤

− n Cβ−1 M n−1 exp{(2δ

n n−1

− α )M



}

n

n

un−1 exp{(α − δ)un−1 }dx 

n n−1

− α )M

}

→0 as  → 0. From above, we can easily get the fact that β → 1, α → λn , r → 0, γ → α as  → 0. Define the rescaling functions v (x) =

u (x + r x) , M 1

w (x) = Mn−1 (u (x + r x) − M ),

(13)

where v (x) and w (x) are defined on  = {x ∈ Rn : x + r x ∈ }. By a direct calculation we obtain that 1

−div(F

n−1

n

n

vn−1 α (un−1 (x +r x)−Mn−1 ) (∇v )Fξ (∇v )) = e + rn γ vn−1 Mn

in  .

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[m1+; v1.304; Prn:27/11/2019; 13:54] P.14 (1-35)

C. Zhou / J. Differential Equations ••• (••••) •••–•••

14 1

Since 0 ≤ v ≤ 1, 1 vn−1 Mn

n n−1

eα (u

n

n

vn−1 α (un−1 (x +r x)−Mn−1 ) Mn e

+ rn γ vn−1 → 0 in Br (0) for any r > 0, and

n

(x +r x)−Mn−1 )

+ rn γ vn−1 is uniformly bounded in L∞ (Br (0)), by Theorem 1

in [29], v is uniformly bounded in C 1,α (B 2r (0)). By Ascoli-Arzela’s theorem, we can find a 1 (Rn ), where v ∈ C 1 (Rn ) and satisfies subsequence j → 0 such that vj → v in Cloc −div(F n−1 (∇v)Fξ (∇v)) = 0

in Rn .

Furthermore, we have 0 ≤ v ≤ 1 and v(0) = 1. The Liouville theorem (see [13]) leads to v ≡ 1. Also we have in  n n−1

1

−div(F n−1 (∇w )Fξ (∇w )) = vn−1 eα (u

n

(x +r x)−Mn−1 )

+ rn M γ un−1  .

(14)

For any r > 0, since 0 ≤ u (x + r x) ≤ M , we have −div(F n−1 (∇w )Fξ (∇w )) = O(1) in Br (0) for small . Then from Theorem 1 in [29] and Ascoli-Arzela’s theorem, there exists 1 (Rn ). Therefore we have w ∈ C 1 (Rn ) such that w converges to w in Cloc n

n

n

n

|u | n−1 (x + r x) − Mn−1 = Mn−1 (vn−1 (x) − 1) n w (x)(1 + O((v (x) − 1)2 )). = n−1

(15)

By taking  → 0, we know that w satisfies n

−div(F n−1 (∇w)Fξ (∇w)) = e n−1 λn w

(16)

in the distributional sense. We also have the facts w(0) = 0 = maxx∈Rn w(x). Moreover, for any R > 0, we have 

n

1 ≥ lim

→0

Wr R (x )



n n−1

n

= lim

vn−1 eα (u

→0



n un−1 α |u | n−1 e dx λ n

(x +r x)−Mn−1 )

dx

WR (0)

=

n

e n−1 λn w dx.

(17)

WR (0)

 n Taking R → +∞, we have Rn e n−1 λn w dx ≤ 1. On the other hand, we claim 

n

e n−1 λn w dx ≥ 1. Rn

(18)

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C. Zhou / J. Differential Equations ••• (••••) •••–•••

15

Actually we can prove it by level-set-method. For t ∈ R, let t = {x ∈ |w(x) > t} and μ(t) = |t |. By the divergence theorem, 

 −div(F

n−1

(∇w)Fξ (∇w))dx =

t

F n−1 (∇w) < Fξ (∇w),

∂t



= ∂t

∇w > ds |∇w|

F n (∇w) ds. |∇w|

By using the isoperimetric inequality (6) and the co-area formula (5), it follows from Hölder inequality that 1



1

nκnn μ(t)1− n ≤ PF (t ) = ∂t



F n (∇w)

≤( ∂t

|∇w|

F (∇w) ds |∇w| 1



ds) n ( ∂t

1 1 ds)1− n |∇w|

 n 1 1 = ( e n−1 λn w dx) n (−μ (t))1− n .

(19)

t

Hence  e

n n−1 λn w

Rn

n dx = λn n−1 n ≤ λn n−1 max  w

= −∞

max  w

n

e n−1 λn t μ(t)dt −∞ max  w

e −∞

n n−1 λn t

−μ (t) 1

n

(nκnn ) n−1

 n 1 ( e n−1 λn w dx) n−1 dt t

  n n n n d ( e n−1 λn w dx) n−1 dt = ( e n−1 λn w dx) n−1 , dt t

Rn

which implies the claim.  n Thus we get that Rn e n−1 λn w dx = 1, which implies that the equality holds in the above isoperimetric inequality. Therefore t must be a wulff ball. In other words, w is radial symmetric with respect to F o (x). We can immediately get w(r) = − where r = F o (x).

1 n n−1 log(1 + κnn−1 r n−1 ) λn

(20)

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C. Zhou / J. Differential Equations ••• (••••) •••–•••

16

Lemma 4.2. For any L > 1, we set u,L = min{u , ML }. Then we have  F n (∇u,L )dx ≤

lim sup →0

Proof. We chose (u −

M + L )



as a test function of (10) to get

 −

1 . L

(u −

M + ) div(F n−1 (∇u )Fξ (∇u ))dx L



 =

1 n M + β un−1 α |u | n−1 (u − e + γ un−1 ) [  ]dx. L λ

(21)



For any R > 0, the estimation of the right hand side of (21) is 

1 n M + β un−1 α |u | n−1 (u − e + γ un−1 ) [  ]dx L λ





1

≥ Wr R (x )



n M + β un−1 α |u | n−1 (u − e dx + o (1) ) L λ

1

= WR (0)

n M + rn β un−1 (x + r x) α |u | n−1 (x +r x) (u (x + r x) − e dx + o (1) ) L λ



n

(v −

= WR (0)



(1 −

→ WR (0)

n 1 n−1 1 + n−1 ) β vn−1 eα (|u| (x +r x)−M ) dx + o (1) L

n 1 )e n−1 λn w dx. L

(22)

In virtue of the divergence theorem and Lemma 2.1, the estimation of the left hand side of (21) is  M + − (u − ) div(F n−1 (∇u )Fξ (∇u ))dx L 

 (u −

=− 

M + M + M + ) div(F n−1 (∇(u − ) )Fξ (∇(u − ) ))dx L L L



F n (∇(u −

= 

M + ) )dx. L

(23)

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[m1+; v1.304; Prn:27/11/2019; 13:54] P.17 (1-35)

C. Zhou / J. Differential Equations ••• (••••) •••–•••

17

Putting (21), (22), (23) together, and taking R → ∞, we obtain  F n (∇(u −

M + 1 ) )dx ≥ 1 − . L L



Noticing that 





F n (∇u )dx = 

F n (∇u,L )dx + 

M + ) )dx. L



Thus the conclusion can be obtained due to the fact Remark 4.3. From Lemma 4.2, applying L 

F n (∇(u −

n−1 n

eλn L|u,L |



F

n (∇u

 )dx

= 1.

2

u,L to inequality (2), we get

n n−1

dx ≤ C < +∞.

(24)

 n n−1

L+1

n n−1

For any L > 1, since eα 2 |u,L | is uniformly bounded in L1 (), then by (2) eα |u,L | q is uniformly bounded in L () for some q > 1. Due to u,L converges to 0 almost everywhere in , it implies that eα |u,L |

n n−1

converges to 1 in L1 (). Thus we have

 lim

→0 {Lu ≤M }

eα |u |

n n−1

 dx = lim

→0

eα |u,L |

n n−1

dx = ||.



Hence  lim

→0 

eα |u | 

= lim

→0 {Lu ≤M }

n n−1

dx

eα |u |

n n−1

→0

≤ || + lim

→0

→0 {Lu >M }



n

≤ || + lim

 dx + lim

λ L n−1 n

Mn−1 λ L

{Lu >M }

eα |u |

n n−1

dx

n n un−1 α |u | n−1 e dx λ

n n−1 n

.

Mn−1

Taking L → 1, we get  lim

→0 

eα |u |

n n−1

dx ≤ || + lim sup →0

λ n

Mn−1

.

(25)

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[m1+; v1.304; Prn:27/11/2019; 13:54] P.18 (1-35)

C. Zhou / J. Differential Equations ••• (••••) •••–•••

18

Noticing that lim→0



α |u | e

n n−1

dx > ||, we have M = 0. →0 λ lim

(26)

The following Lemma will be used in Section 5. Lemma 4.4.





n

lim

e

→0

α |u | n−1

eα |u |

dx = || + lim lim sup R→+∞

→0



n n−1

dx.

WRr (x )

Proof. On one hand, 

eα |u |

lim sup →0

n n−1

dx

WRr (x )



eα |u |

≤ lim sup →0



n n−1

→0





≤ lim sup →0

e

eα |u |

dx − lim inf

n α |u | n−1

n n−1

dx

\WRr (x )

dx − ||.

(27)



On the other hand, 

eα |u |

n n−1

λ

dx =

n n−1

M

WRr (x )



n

e n−1 λn w dx + o (1)),

( WR (0)

which gives  R→+∞

→0

n n−1

eα |u |

lim lim sup

λ

dx = lim sup

n

WRr (x )

Combining (25), (27), (28) and Lemma 3.2, we get the result.

(28)

.

Mn−1

→0

2

Now we claim that  lim

→0

n 1 M n−1 β un−1 eα |u | dx = 1. λ

(29)



To this purpose, we denote ϕ = 

 ϕ dx =



1

{Lu
n

M n−1 α |u | n−1 λ β u e

. Clearly

 ϕ dx + {Lu ≥M }\Wr R (x )

 ϕ dx + Wr R (x )

ϕ dx.

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[m1+; v1.304; Prn:27/11/2019; 13:54] P.19 (1-35)

C. Zhou / J. Differential Equations ••• (••••) •••–•••

19

We estimate the three integrals on the right hand side respectively. By (26) and Lemma 4.1 we have   n 1 M β n−1 0≤ ϕ dx = un−1 eα |u | dx λ {Lu
{Lu
M β ≤ λ



1

n−1 α |u,L | u,L e

n n−1

dx



1 = o (1)O( ). L

(30)

Moreover for any R > 0, we have 

 ϕ dx ≤

{Lu ≥M }\Wr R (x )

{Lu ≥M }\Wr R (x )



≤ \Wr R (x )

n n Lβ n−1 n−1 u eα |u | dx λ

n n Lβ n−1 n−1 u eα |u | dx λ



n

e n−1 λn w dx),

→ L(1 −

(31)

WR (0)

and 

 ϕ dx =

Wr R (x )

1

β vn−1 eα (|u |

n n n−1 −Mn−1 )

dx

WR (0)



→

n

e n−1 λn w dx.

(32)

WR (0)

Putting (30) (31) (32) together and taking  → 0 first, then letting R → ∞, we conclude (29). In the similar way, we also can obtain that  lim

→0

n 1 β M n−1 n−1 u eα |u | φ(x)dx = φ(x0 ) λ

(33)



for any φ(x) ∈ Cc0 (). The following phenomenon was first discovered by Brezis and Merle [3], developed later by Struwe [23] and Yang [35,36]. We deduce the new version involving n-Finsler-Laplacian. Lemma 4.5. Let {f } be a uniformly bounded sequence of functions in L1(), and {ψ } ⊂ C 1 () ∩ W01,n () satisfy −div(F n−1 (∇ψ )Fξ (∇ψ )) = f + αψ |ψ |n−2

in ,

(34)

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[m1+; v1.304; Prn:27/11/2019; 13:54] P.20 (1-35)

C. Zhou / J. Differential Equations ••• (••••) •••–•••

20

where 0 ≤ α < λ1 () is a constant. Then for any 1 < q < n, we have ||∇ψ ||Lq () ≤ C for some constant C depending only on q, n,  and the upper bound of ||f ||L1 () . Proof. When α = 0. We use an argument of M. Struwe to prove that ||∇ψ ||Lq () ≤ C||f ||L1 () for some constant C depending only on q, n, . Without loss of generality, we assume t + + ||f ||L1 () = 1. For t ≥ 1, denote  ψ = min{ψ , t},  where ψ is a positive part of ψ . Testing Eq. (34) with ψt , we have  F n (∇ψt )dx ≤  |f |ψt ≤ t . Assume || = |Wd |, where Wd = {x ∈ Rn : F o (x) ≤ d}. Let ψ be the nonincreasing rearrangement of ψt , and |Wρ | = |{x ∈ Wd : ψ ≥ t}|. It is known that ||F (∇ψ )||Ln (Wd ) ≤ ||F (∇ψt )||Ln () , and we have  φ∈W01,n (Wd ),φ|Wρ =t

 F n (∇φ)dx ≤

inf

Wd

F n (∇ψ )dx ≤ t.

(35)

Wd

The above infimum can be attained by  φ1 (x) =

t log F od(x) / log ρd t

in Wd \Wρ , in Wρ .

Calculating ||F (∇φ1 )||nLn (Wd ) , we have by (35), ρ ≤ de−C1 t for some constant C1 > 0. Hence |{x ∈  : ψ ≥ t}| = |Wρ | ≤ κn d n e−nC1 t . For any 0 < δ < nC1 , 

+

eδψ dx ≤ eδ || +

∞ 

e(m+1)δ |{x ∈  : m ≤ ψ ≤ m + 1}|

m=1



≤ eδ || + κn d n eδ

∞ 

e−(nC1 −δ)m ≤ C2

m=1

for some constant C2 . Testing Eq. (34) with log  

1+2ψ+ , 1+ψ+

we have

F n (∇ψ+ ) dx ≤ log 2. (1 + ψ+ )(1 + 2ψ+ )

By the Young inequality, we have for any 1 < q < n,  



 q F n (∇ψ+ ) ((1 + ψ+ )(1 + 2ψ+ )) n−q dx dx + + + (1 + ψ )(1 + 2ψ )    + ≤ C3 (1 + eδψ dx) ≤ C4 ,

F q (∇ψ+ )dx ≤



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[m1+; v1.304; Prn:27/11/2019; 13:54] P.21 (1-35)

C. Zhou / J. Differential Equations ••• (••••) •••–•••

21

for some constants C3 and C4 depending only on q, n and . Let ψ− be the negative part of ψ . Similarly, we have  F q (∇ψ− )dx ≤ C5 for some constant C5 depending only on q, n and . Then by Lemma 2.1, the lemma holds. When α ∈ (0, λ1 ()). Suppose ψ is unbounded in Ln−1 (). Then there exists a subsequence {j } such that ||ψj ||Ln−1 () → +∞ as j → +∞. Let wj = ψj /||ψj ||Ln−1 () . Then we have 1,q

||wj ||Ln−1 () = 1, and −Qn (wj ) is bounded in L1 (). Hence wj is bounded in W0 () for 1,q

any 0 < q < n. Assume wj converges to w weakly in W0 () and strongly in Ln−1 (). It can be easily derived that w is a weak solution of −Qn u = αw|w|n−2 in . Since 0 < α < λ1 (), w must be zero. On the other hand, ||wj ||Ln−1 () = 1 leads to ||w||Ln−1 () = 1, contradiction. Therefore ψ must be bounded in Ln−1 (). It implies f + αψ |ψ |n−2 ∈ L1 (). Then, for any 1 < q < n, there exist a constant C depending only on q, n, , and the upper bounded of ||f ||L1 () such that ||∇ψ ||Lq () ≤ C. Thus the proof is finished. 2 The following lemma reveals how u converges away from x0 . 1

1,q

Lemma 4.6. Mn−1 u  Gα weakly in W0 () for any 1 < q < n, where Gα is a Green function satisfying 

−div(F n−1 (∇Gα )Fξ (∇Gα )) = δx0 + αGn−1 α Gα = 0

in , on ∂.

(36)

1

Furthermore, Mn−1 u → Gα in C 1 ( ) for any domain  ⊂⊂ \{x0 }. Proof. By Eq. (10), we have 1 1 n−1

−Qn (M

n M β un−1 α |u | n−1 u ) = e + γ M un−1  . λ 1

(37) 1,q

Due to (29) and Lemma 4.5, we obtain that Mn−1 u is uniformly bounded in W0 () for any 1

1 < q < n. Assume Mn−1 u  Gα weakly in W0 (). Testing Eq. (37) with φ ∈ C0∞ (), we have by (33)  −

1,q

1 n−1

φQn (M 

 u )dx =

1 n M β un−1 α |u | n−1 φ e dx + γ λ



φM un−1  dx 



→ φ(x0 ) + α



φGn−1 α dx. 

Hence  ∇φF n−1 (∇Gα )Fξ (∇Gα )dx = φ(x0 ) + αGn−1 α , 

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[m1+; v1.304; Prn:27/11/2019; 13:54] P.22 (1-35)

C. Zhou / J. Differential Equations ••• (••••) •••–•••

22

in the sense of measure and whence (36) holds. For any fixed small δ, we choose a cut-off function ξ(x) ∈ C0∞ (\Wδ (x0 )) such that  n ξ(x) = 1 on \W3δ (x0 ). By Lemma 4.1 we get  F (∇(ξ u ))dx → 0 as  → 0. Then e(ξ u )

n n−1

n n−1

is bounded in Ls (\Wδ (x0 )) for any s > 1. In particular, eu 1 n−1

Ls (\W3δ (x0 )). Since M 1 M un−1 λ

is bounded in

u is bounded in Lq () for any q > 1, Hölder inequality im-

n α |u | n−1

plies that e is uniformly bounded in Ls0 (\W3δ (x0 )). From the proof of the ∈ L1 (\W3δ (x0 )). Then by Lemma 2.2, Lemma 4.5 and 0 ≤ γ < λ1 (), we have γ M un−1  we have 1

||Mn−1 u ||L∞ (\W3δ (x0 )) < C. 1

Theorem 1 in [29] and Ascoli-Arzela’s theorem, we have Mn−1 u converges to Gα in 1 Cloc (\W4δ (x0 )). 2 Lemma 4.7. Asymptotic representation of Green function Gα is 1

Gα = −

1

log F o (x − x0 ) + CG + ψ(x)

(38)

(nκn ) n−1 1 (\{x }) such that lim o where CG is a constant, ψ(x0 ) = 0 and ψ(x) ∈ C 0 () ∩ Cloc 0 x→x0 F (x − x0 )∇ψ(x) = 0.

Up to now, we have described the convergence behavior of u near x0 and away from x0 when the concentration point x0 in the interior of . Case 2. x0 lies on ∂. Lemma 4.8. Let d = dist (x , ∂), and r be defined in (12). There holds r /d → 0. Proof. Suppose not, there exists R > 0 such that d ≤ Rr . Take some y ∈ ∂ such that d = |x − y |. Let v  = M−1 u (y + r x). By a reflection argument, similar to the case 1, we have v  → 1 in C 1 (BR+ ) for ||v  ||L∞ (B + ) = 1. This contradicts v  (0) = 0. 2 R

Set  = {x ∈ Rn |x + r x ∈ }. By Lemma 4.8, we have lim→0 dist (xr ,∂) → +∞, then  → Rn . Let w be define in (13) and 1 (Rn ). We proceed w be define in (20). Similar arguments to Case 1 imply that w → w in Cloc 1

as in Case 1, Mn−1 u  Gα weakly in W01,n (), and in C 1 (), where G satisfies the following equation: 

−Qn Gα = 0 Gα = 0

in , in ∂.

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C. Zhou / J. Differential Equations ••• (••••) •••–•••

23

The above equation has a unique solution G = 0. Hence 1

1

weakly in W01,n (),

Mn−1 u  0

Mn−1 u → 0

in C 1 (\{x0 }).

(39)

This is all we need to know about the convergence behavior of u when the concentration point x0 lies on the boundary of . The proof point (1) of Theorem 1.1. If M is bounded, elliptic estimates implies that the Theorem holds. If M → +∞, then we have ||u ||Ln () → 0. A straightforward calculation gives  Jλαn − (u ) =

1

n

e

n

(λn −)|u | n−1 ((1+α||u ||nLn () ) n−1 −1) (λn −)|u | n−1

e

dx

 n

≤e

1

λn Mn−1 ((1+α||u ||nLn () ) n−1 −1)



eλn |u |

n n−1

dx



=e

1 1 − n λn α n−1 u ||nLn () +M n−1 O(||Mn−1 u ||2n ) n−1 ||M Ln ()



eλn |u |

n n−1

dx.

 1

Notice that α satisfies 0 ≤ α < λ1 (). When x0 ∈ , we have ||Mn−1 u ||Ln () → ||Gα ||nLn () , 1

when x0 ∈ ∂, ||Mn−1 u ||Ln () → 0. Hence, together with Lemma 3.2 and (2) completes the proof of point (1) of Theorem 1.1. 2 5. Proof of Theorem 1.2 In this section, we will prove our main Theorem. We first give a Lemma in [43]. Lemma 5.1. Assume that u is a normalized concentrating sequence in W01,n (Wρ ) with a blow up point at the origin, i.e.  (1) Wρ F n (∇u )dx = 1, 1,n (2) u  0 weakly in W  0 (Wρ ),n (3) for any 0 < r < ρ, Wρ \Wr F (∇u )dx → 0. Then  lim sup →0

(eλn |u |

n n−1

1

1

− 1)dx ≤ κn ρ n e1+ 2 +···+ n−1 .

(40)

Wρ

Motivated by the arguments in [6,33,35,37,41], we first compute an upper bound of T0 if u blows up.

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24

Lemma 5.2. If lim sup→0 ||u ||∞ = ∞, then 1

1

sup Jλαn (u) ≤ || + κn eλn CG +1+ 2 +···+ n−1 .

(41)

u∈H

Proof. Case 1. x0 lies in the interior of . Note that x0 is a blow-up point of uε . We chose Wδ (x0 ) ⊂  for sufficient small δ > 0. By (36) we have  F n (∇Gα )dx \Wδ (x0 )





=

Gα F

n−1

(∇Gα )Fξ (∇Gα ), νds −

∂(\Wδ (x0 ))

\Wδ (x0 )



+α

Gα div(F n−1 (∇Gα )Fξ (∇Gα ))dx

|Gα |n dx

\Wδ (x0 )



=−



Gα F

n−1

(∇Gα )Fξ (∇Gα ), νds + α

∂ Wδ (x0 )

|Gα |n dx.

(42)

\Wδ (x0 )

Due to (38) and Lemma 2.1, we have on ∂Wδ (x0 ) 1

F (∇Gα ) = F (−

1

∇F o (x − x0 ) 1 + o( o )) o F (x − x0 ) F (x − x0 )

(nκn ) n−1 1 1 + o( ), = 1 δ (nκn ) n−1 δ

(43)

and Fξ (∇Gα ), ν = Fξ (∇Gα ),

∇F o (x − x0 )  |∇F o (x − x0 )|

= Fξ (∇Gα ), (−(nκn )

1 n−1

F (x − x0 )) o

1 ∇Gα − o( F o (x−x ) 0)

|∇F o (x − x0 )|



o( 1δ ) F (∇Gα ) − ) |∇F o (x − x0 )| |∇F o (x − x0 )| 1 , = −(1 + oδ (1)) o |∇F (x − x0 )| 1

= −(nκn ) n−1 δ(

where oδ (1) → 0 as δ → 0. Putting (38), (43), (44) into (42), we obtain   1 n F (∇Gα )dx = − log δ + CG + α |Gα |n dx + oδ (1) 1 (nκn ) n−1 \Wδ (x0 )

\Wδ (x0 )

(44)

(45)

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Hence from Lemma 4.6 we have  1 1 F n (∇u )dx = log δ + CG + α n (− 1 (nκn ) n−1 Mn−1

\Wδ (x0 )

25

 |Gα |n dx + oδ (1) + o (1)),

\Wδ (x0 )

(46) where o (1) → 0 as  → 0. Next we let b = sup∂ Wδ (x0 ) u and u = (u − b )+ . Then u ∈ W01,n (Wδ (x0 )). From (46) and the fact that Wδ (x0 ) F n (∇u )dx = 1 − \Wδ (x0 ) F n (∇u )dx, we have 

1

F n (∇u )dx = τ ≤ 1 −

n n−1

(−

M 

Wδ (x0 )

+α

1 1

log δ + CG

(nκn ) n−1 |Gα |n dx + oδ (1) + o (1)).

\Wδ (x0 )

By Lemma 5.1, 

n 1/n n−1 |

(eλn |u /τ

lim sup →0

1

1

− 1)dx ≤ κn δ n e1+ 2 +···+ n−1 .

Wδ (x0 )

Now we focus on the estimate in the bubbling domain WRr (x ). According to the rescaling 1 (Rn ), and whence u = M + o (R), functions in Section 4, we can assume that w → w in Cloc    where o (R) → 0 as  → 0 for any fixed R > 0. Then from Lemma 4.6 we have n

n

n

α |u | n−1 ≤ λn (1 + α||u ||nLn () ) n−1 (u + b ) n−1 n

≤ λn un−1 +

1 λn α n ||Gα ||nLn () + α b un−1 + o (1), n−1 n−1

(47)

and 1

b un−1 = −

1 1

log δ + CG + oδ (1) + o (1).

(48)

(nκn ) n−1 Notice that n n n−1

λn u

≤

λn un−1 1 n−1

−

λn 1 log δ + α||Gα ||nLn () + CG + oδ (1) + o (1)) (− 1 n − 1 (nκn ) n−1

+

n λn αλn log δ − ||Gα ||nLn () − CG + oδ (1) + o (1). n−1 n−1 n−1

τ

n

=

λn un−1 1 n−1

τ

Combining (47)-(49), we obtain in WRr (x )

(49)

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26

n

α |u |

n n−1

λn un−1

≤

1 n−1

− n log δ + λn CG + oδ (1) + o (1).

τ Therefore, we have 

eα |u |

lim sup →0

n n−1

dx

WRr (x )



≤ δ −n eλn CG +oδ (1) lim sup →0

≤δ

−n λn CG +oδ (1)

e

− 1)dx

WRr (x )



lim sup →0

n 1/n n−1 |

(eλn |u /τ

n 1/n n−1 |

(eλn |u /τ

− 1)dx

Wδ (0) 1

1

≤ δ −n eλn CG +oδ (1) κn δ n e1+ 2 +···+ n−1 . Taking δ → 0, we have  lim sup →0

eα |u |

n n−1

1

1

dx ≤ κn eλn CG +1+ 2 +···+ n−1 .

WRr (x )

Then by the Lemma 4.4, we obtain  n 1 1 n−1 lim sup eα |u | dx ≤ || + κn eλn CG +1+ 2 +···+ n−1 . →0

 1

1

It follows Lemma 3.2 to get supu∈H Jλαn (u) ≤ || + κn eλn CG +1+ 2 +···+ n−1 . Case 2. x0 lies on the boundary of . 1

1,q

We proceed and use the same notions as in case 1. By (39), Mn−1 u  0 weakly in W0 () for any 1 < q < n, and in C 1 (\{x0 }). Hence  o (1) F n (∇u )dx ≤ τ = 1 − n/(n−1) , (50) M Wδ (x )

and we have in ωRr (x ), n

n

α |u | n−1 ≤ λn |u /τ1/n | n−1 + o (1). Combining (50), (51) and Lemma 4.4, we have  n 1 1 n−1 lim sup eα |u | dx ≤ || + O(δ n )e1+ 2 +···+ n−1 . →0



(51)

(52)

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27

Letting δ → 0, (52) together with Lemma 3.2 gives supu∈H Jλαn (u) ≤ ||, which is impossible. Therefore we conclude that x0 cannot lie on ∂. 2 In Lemma 5.2 we have got the upper bound of supu∈H Jλαn (u) if u blows up. Next we will construct an explicit test function to get the lower bound of supu∈H Jλαn (u), which will contradict the upper bound of supu∈H Jλαn (u). Thus we get a contradiction and consequently we complete the proof of Theorem. The similar arguments can be seen in [37,39,41]. Lemma 5.3. There holds 1

1

sup Jλαn (u) > || + κn eλn CG +1+ 2 +···+ n−1 .

(53)

u∈H

Proof. Define a sequence of functions in  by

φ =

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

C+C

1 − n−1

C

1

n F o (x−x0 ) n−1 ) )+b)  −n 1 n n−1 n (1+αC ||Gα ||Ln () )

n−1 (− n−1 ( λn log(1+κn

1 − n−1 −n

(G−ηψ) 1

(1+αC n−1 ||Gα ||nLn () ) n − 1 C n−1 G −n 1 (1+αC n−1 ||Gα ||nLn () ) n

, x ∈ WR (x0 ),

,

x ∈ W2R (x0 )\WR (x0 ),

,

x ∈ \W2R (x0 ),

where G and ψ are functions given in (38), R = − log , η ∈ C01 (W2R (x0 )) satisfying that 2 η = 1 on WR (x0 ) and |∇η| ≤ R , b and C are constants depending only on  to be determined later. Clearly W2R (x0 ) ⊂  provided that  is sufficiently small. In order to assure that φ ∈ W01,n (), we set 1

C + C − n−1 (−

1 n 1 n−1 1 log(1 + κnn−1 R n−1 ) + b) = C − n−1 (− log(R) + CG ), 1 λn (nκn ) n−1

which gives 1

n

C n−1 = −

(nκn )

1 n−1

log  +

n 1 log κn − b + CG + O(R − n−1 ). λn

 Next we make sure that  F n (∇φ )dx = 1. By the coarea formula (6), we have  WR (x0 )

(F

o (x−x ) 0

1 n−1

(1 + κn



R

n

) n−1 1n

n o 0 ) n−1 n ( F (x−x ) ) 

n

( s ) n−1 1n

dx = nκn

1 n−1

0

(1 + κn 1

=

n−1 κn

0

s n−1 ds

n

κnn−1  R n−1

1 n−1

n ( s ) n−1 )n

t n−1 dt. (1 + t)n

(54)

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28

Then it follows that 1



n−1

F n (∇φ )dx =

λn (C

WR (x0 )

n n−1

n

κnn−1  R n−1

+ α||Gα ||nLn () )

0 1

λn (C

n n−1

+ α||Gα ||nLn () ) n−1

=

λn (C

n n−1

+ α||Gα ||nLn () 1 n−1

+ log(1 + κn

R

n n−1

λn (C

n n−1

k=0

n

) + O(R − n−1 ))

+ α||Gα ||nLn () ) 1

0

(t + 1 − 1)n−1 dt (1 + t)n

n−2 k  Cn−1 (−1)n−1−k ( n−k−1 )

n−1

=

n

κnn−1  R n−1

n−1

=

t n−1 dt (1 + t)n

(−(1 +

n

1 1 + ··· + ) 2 n−1

n

+ log(1 + κnn−1 R n−1 ) + O(R − n−1 )),

(55)

where we have used the fact that −

n−2 k  Cn−1 (−1)n−1−k

n−k−1

k=0

=1+

1 1 + ··· + . 2 n−1

1 Noting that ψ(x) satisfies that |∇ψ(x)| = o( F o (x−x ) as x → x0 , and using Lemma 2.1 and 0) (38), we have

 F n (∇Gα ) − F n (∇(Gα − ηψ))dx = oR (1). W2R (x0 )\WR (x0 )

Then together with (45), we have  F (∇φ )dx = \WR (x0 )



1

n

n

C n−1 + α||Gα ||nLn ()  −

F n (∇Gα )dx

( \WR (x0 )

F n (∇Gα ) − F n (∇(Gα − ηψ))dx)

W2R (x0 )\WR (x0 )

=

1 C +

n n−1

+ α||Gα ||nLn ()

(−

1 1

log(R)

(nκn ) n−1

αλn ||Gα ||nLn () + CG + oR (1)). n−1

(56)

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29

Putting (55), (56) together, we have 

n−1

F n (∇φ )dx =

λn (C



+ Since



F

n (∇φ

 )dx

n n−1

+ α||Gα ||nLn () )

(−

λn 1 n log  + log κn + CG n−1 n−1 n−1

n αλn 1 1 ||Gα ||nLn () − (1 + + · · · + ) + O(R − n−1 ) + oR (1)). n−1 2 n−1

= 1, we have n

C n−1 =

n−1 n λn 1 (− log  + log κn + CG λn n−1 n−1 n−1 n 1 1 −(1 + + · · · + ) + O(R − n−1 ) + oR (1). 2 n−1

(57)

Consequently from (54), we have b=

n (n − 1) 1 1 (1 + + · · · + ) + O(R − n−1 ) + oR (1). λn 2 n−1

(58)

Since n2

||φ ||nLn () =

||Gα ||nLn () + O(C n−1 R n  n ) + O((Rε)n (log(R)n )) n

C n−1 + α||Gα ||nLn ()

,

using the inequality 1

(1 + t) n+1 ≥ 1 −

t , for t small . n−1

In view of (57) and (58), there holds in WR (x0 ), 1

n

λn |φ (x)| n−1 (1 + α||φ ||nLn () ) n−1 1

n

≥ λn C n−1 − n log(1 + κnn−1 (

α 2 λn ||Gα ||2n F o (x − x0 ) n nλn Ln () ) n−1 ) + b− n  n−1 (n − 1)C n−1

n2

2n

+O(C − n−1 ) + O(C n−1 (R)n ) + O((R)n (− log(R))n ) 1 1 ) ≥ −n log  + log κn + λn CG + (1 + + · · · + 2 n−1 1

−n log(1 + κnn−1 ( 2n

α 2 λn ||Gα ||2n F o (x − x0 ) n Ln () ) n−1 ) − n  (n − 1)C n−1 −n

+O(C − n−1 ) + O(R n−1 ) + oR (1), n

where we have used the inequality |1 + t| n−1 ≥ 1 +

n n−1 t

+ O(t 3 ) for small t . By using the fact

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30

n−2 k  Cn−2 (−1)n−k−2

n−k−1

k=0

=

1 , n−1

one can get 

1 n o n−1 F (x−x0 ) n−1 ( ) ) 

e−n log −n log(1+κn

dx

WR (x0 )

=

1 n



1 1 n−1

(1 + κn

WR (x0 )

1

(F

o (x−x ) 0



dx

n

) n−1 )n

n

κnn−1  R n−1

= (n − 1) 0 1 n κnn−1 R n−1



= (n − 1) 0

t n−2 dt (1 + t)n

(t + 1 − 1)n−2 dt (1 + t)n

n n 1 ≥ (n − 1)( + O(R − n−1 )) = 1 + O(R − n−1 )). n−1

Then we obtain 

1

n

e

λn |φ (x)| n−1 (1+α||φ (x)||nLn () ) n−1

1

1

dx ≥ κn eλn CG +(1+ 2 +···+ n−1 ) (1 −

WR (x0 ) 2n

n

α 2 λn ||Gα ||2n Ln ()

+O(C − n−1 ) + O(R − n−1 ) + oR (1). On the other hand, since 

n

n

|Gα | n−1 dx = O((R)n log n−1 (R)) = oR (1), W2R (x0 )

we obtain 

1

n

e

λn |φ (x)| n−1 (1+α||φ (x)||nLn () ) n−1

dx

\WR (x0 )



n

≥

(1 + λn |φ (x)| n−1 )dx

\W2R (x0 ) n

≥ || +

λn ||Gα || n−1n

L n−1

C

n (n−1)2

−2n

+ O(C (n−1)2 ) + oR (1)

n

(n − 1)C n−1

)

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31

Together with the above integral estimates on WR , and the fact that −2n

−n

C (n−1)2 → 0 and R (n−1) → 0 as  → 0. Then we have 

1

n

e

λn |φ (x)| n−1 (1+α||φ (x)||nLn () ) n−1

1

1

dx > || + κn eλn CG +(1+ 2 +···+ n−1 ) ,



provided that  > 0 is chosen sufficiently small. Thus we get the conclusion of Lemma. 2 6. Asymptotic representation of Gα In this section we will give the asymptotic representation of Green function Gα , similarly to [15,32,35] n

The proof of Lemma 4.7. Since ckn−1 uk ≥ 0 in \{0}, we have Gα ≥ 0 in \{0}. Theorem 1 in [24] gives 1 Gα ≤ ≤K K − log r

in \{0}

(59) 1

α (rk x) for some constant k > 0. Assume (r) = −c(n) log r, c(n) = (nκn )− n−1 . Let Gk = G(r , k) which is defined in {x ∈ Rn \{0}, rk x ∈ Wδ } for some small δ > 0. Here rk → 0 as k → +∞. Then Gk satisfy the equation

−

n  ∂ (F n−1 (∇Gk )Fξ (∇Gk )) = αrkn Gn−1 k . ∂xi i=1

1 (Rn \{0}) and G∗ is bounded, By theorem 1 in [29], when rk → 0, Gk converges to G∗ in Cloc ∗ where G satisfying

−

n  ∂ (F n−1 (∇G∗ )Fξ (∇G∗ )) = 0. ∂xi i=1

From serrin’s result (see [25]) and (59), 0 is a removable singularity and G∗ can be extended to ˆ must be a constant. Let ˆ ∈ C 1 (Rn ). Consequently, form Liouville type theorem (see [13]), G G Gα (x) ˆ =γ. γk = supWδ \Wr (x) , and γ = limk→+∞ γk , (γ > 0). This means the constant function G k Set o G+ η (x) = (γ + η)((x) − (δ)) − c(n)(γ + η)(F (x) − δ) + sup Gα ,

(60)

o G− η (x) = (γ − η)((x) − (δ)) − c(n)(γ − η)(F (x) − δ) + inf Gα .

(61)

∂ Wδ

∂ Wδ

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32

A straightforward calculation shows n−1 1 ( o + 1)n−2 , o F (x) F (x) n−1 1 n−1 (n)(γ − η)n−1 o ( o − 1)n−2 . −Qn G− η (x) = c F (x) F (x)

n−1 −Qn G+ (n)(γ + η)n−1 η (x) = c

(62) (63)

By, for any fixed 0 < η < γ , we have −Qn G+ η (x) ≥ −Qn G G+ η |∂ Wδ ≥ Gα |∂ Wδ ,

in

Wδ \Wrk ,

G+ η |∂ Wrk ≥ Gα |∂ Wrk ,

provided that δ are sufficiently small and rk < δ. By the comparison principle (see [34]), we have Gα ≤ (γ + η)(x) + Cδ

in

Wδ \Wrk

(64)

for some constant Cδ . Letting η → 0 first, then k → ∞, one has Gα ≤ γ (x) + Cδ

in

Wδ \{0}.

A similar argument gives Gα ≥ γ (x) + Cδ in Wδ \{0} for some constant Cδ . Hence Gα − γ (x) is bounded in L∞ (Wδ ). Next we prove the continuity of Gα − γ (x) at 0. We look at the points where the bounded function Gα − γ (x) achieves its supremum in Wδ . Set λ = supWδ (Gα − γ (x)). λ achieves at some point in Wδ \{0}, then Gα − γ (x) − γ c(n)F o (x) also achieves at some point in Wδ \{0}. It follows from comparison principle (see [8]) that Gα − γ (x) − γ c(n)F o (x) is a constant, hence we have done. λ achieves at 0, set wr (x) = Gα (rx) − γ (r)

in

W δ \{0}. r

n n−1 The function wr satisfies −Qn (wr (x)) − αr n Gn−1 α (rx) = 0. We also have r Gα (rx) ∈ L∞ (Wδ ) and |wr − γ (x)| ≤ C0 for C0 = supWδ \{0} |Gα − γ (x)|. By Theorem 1 in [29], 1 (Rn \{0}), where w ∈ C 1 (Rn \{0}) satisfies −Q (w) = 0. For the when r → 0, wr → w in Cloc n

sequence ξj =

xrj rj

, F o (ξj ) = 1, which maybe assumed to converge to ξ 0 ∈ ∂W1 , we have wrj (ξj ) − γ (ξj ) = Gα (xrj ) − γ (xrj ) → λ.

Hence w(x) ≤ γ (x) + λ

and

w(ξ 0 ) = γ (ξ 0 ) + λ.

By comparison principle (see [34]), w(x) = γ (x) + λ and hence wr → γ (x) + λ in 1 (Rn \{0}). This implies Cloc

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lim (Gα (rx) − γ (rx)) = λ,

lim ∇x (Gα (rx) − (rx)) = 0.

r→0

r→0

33

(65)

The above equalities lead to the continuity of Gα − γ  and limx→0 F 0 (x)∇(Gα − γ ) = 0. We assume supx∈Wδ (Gα − γ ) = supF o (x)=δ (Gα − γ ), we define wr as the above, then 1 (Rn \{0}) and |w − γ | ≤ C . We now look at the points where w − γ  achieves wr → w in Cloc 0 its supremum in Rn . Set λ˜ = supRn (w − γ ). If λ˜ is achieved at some point in Rn \{0}, then w − γ  equals to some constant by strong 1 (Rn \{0}) as r → 0. For maximum principle [12], which implies Gα (rx) − γ (rx) → λ˜ in Cloc any fixed  > 0, there exists n0 such that n ≥ n0 and x ∈ ∂W1 , we have γ (rn x) + λ˜ −  ≤ Gα (rn x) ≤ γ (rn x) + λ˜ + . Applying maximum principle [12] in Wrn0 \Wrn we obtain γ (x) + λ˜ −  ≤ Gα (x) ≤ γ (x) + λ˜ + , which leads to with λ replaced by λ˜ . If λ˜ is achieved at 0, we simply argue as in the above to deduce lim (w − γ ) = λ˜

x→0

and hence

˜ lim lim (Gα (rn x) − γ (rn x)) = λ.

x→0 rn →0

(66)

If λ˜ is achieved at ∞, the same idea as in case can be applied when we defined λ(R) = maxδ≤F o (x)≤R (w − γ ) = max∂ WR (w − γ ) and let R tend to ∞. We obtain ˜ lim (w − γ ) = λ,

x→∞

˜ lim lim (Gα (rn x) − γ (rn x)) = λ.

x→∞ rn →0

(67)

As long as we have (66)and (67), we can have use maximum principle [12] again to conclude (65) as before. Integrating by parts on both sides of Eq. (36) over Wδ , we have  − Wδ

 div(F n−1 (∇Gα )Fξ (∇Gα ))dx+ = 1 + α

Gn−1 α dx.

(68)

Wδ

Because Gα (x) = γ (x) + o(1) and ∇Gα (x) = γ ∇(x) + o( F o1(x) ) as x → 0. Inserting the above two equalities into (68), then letting δ → 0, we obtain γ = 1. 2 References [1] Adimurthi, O. Druet, Blow-up analysis in dimension 2 and aharp form of Trudinger-Moser inequality, Commun. Partial Differ. Equ. 29 (2004) 295–322. [2] A. Alvino, V. Ferone, G. Trombetti, P. Lions, Convex symmetrization and applications, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 14 (1997) 275–293. [3] H. Brezis, F. Merle, Uniform estimates and blow-up behavior for solutions of −u = V (x)eu in two dimensions, Commun. Partial Differ. Equ. 16 (1991) 1223–1253. [4] M. Belloni, V. Ferone, B. Kawohl, Isoperimetric inequalities, wulffshape and related questions for strongly nonlinear elliptic operators, Z. Angew. Math. Phys. 54 (2003) 771–783.

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