A sharp Trudinger-Moser type inequality involving L^{n} norm in the entire space \mathbb{R}^{n}
aa r X i v : . [ m a t h . A P ] M a r A sharp Trudinger-Moser type inequality involving L n norm in the entire space R n Guozhen LuDepartment of MathematicsUniversity of ConnecticutStorrs, CT 06269, USAE-mail: [email protected] ZhuFaculty of ScienceJiangsu UniversityZhenjiang, 212013, ChinaE-mail: [email protected]
Abstract
Let W ,n ( R n ) be the standard Sobolev space and k·k n be the L n norm on R n . Weestablish a sharp form of the following Trudinger-Moser inequality involving the L n norm sup k u k W ,n ( R n ) =1 ˆ R n Φ (cid:16) α n | u | nn − (1 + α k u k nn ) n − (cid:17) dx < + ∞ in the entire space R n for any 0 ≤ α <
1, where Φ ( t ) = e t − n − P j =0 t j j ! , α n = nω n − n − and ω n − is the n − R n . We also showthat the above supremum is infinity for all α ≥
1. Moreover, we prove the supremumis attained, namely, there exists a maximizer for the above supremum when α > Mathematics Subject Classification . Primary 35J50; Secondary 35J20,46E35.
Key words and phrases: Trudinger-Moser inequality; Blow up analysis; Extremal functions; unboundeddomain .The first author was partially supported by a US NSF grant and a Simons fellowship from Simons founda-tion and and the second author was partially supported by Natural Science Foundation of China (11601190),Natural Science Foundation of Jiangsu Province (BK20160483) and Jiangsu University Foundation Grant(16JDG043). G. Lu and M. Zhu is sufficiently small. The proof is based on the method of blow-up analysis of thenonlinear Euler-Lagrange equations of the Trudinger-Moser functionals.Our result sharpens the recent work [12] in which they show that the above inequal-ity holds in a weaker form when Φ( t ) is replaced by a strictly smaller Φ ∗ ( t ) = e t − n − P j =0 t j j ! .(Note that Φ( t ) = Φ ∗ ( t ) + t n − ( n − ). Let Ω ⊆ R n be an open set and W ,q (Ω) be the usual Sobolev space, that is, the completionof C ∞ (Ω) under the norm k u k W ,q (Ω) = (cid:18) ˆ Ω ( | u | q + |∇ u | q ) dx (cid:19) q . If 1 ≤ q < n , the classical Sobolev embedding says that W ,q (Ω) ֒ → L s (Ω) for 1 ≤ s ≤ q ∗ ,where q ∗ := nqn − q . When q = n , it is known that W ,n (Ω) ֒ → L s (Ω) for any n ≤ s < + ∞ ,but W ,n (Ω) L ∞ (Ω). The analogue of the optimal Sobolev embedding is the well-knownTrudinger-Moser inequality ([23],[28]) which states as follows(1.1) sup u ∈ W ,n (Ω) k∇ u k Ln (Ω) ≤ ˆ Ω e α | u | nn − dx < ∞ iff α ≤ α n = nω n − n − , where ω n − is the n − R n and Ω is a domainof finite measure in R n .Due to a wide range of applications in geometric analysis and partial differential equa-tions (see [7], [4], [14] and references therein), numerous generalizations, extensions andapplications of the Trudinger-Moser inequality have been given. We recall in particularthe result obtained by P.-L. Lions [19], which says that if { u k } is a sequence of functionsin W ,n (Ω) with k∇ u k k L n (Ω) = 1 such that u k → u weakly in W ,n (Ω), then for any0 < p < (cid:16) − k∇ u k nL n (Ω) (cid:17) − / ( n − , one hassup k ˆ Ω e α n p | u k | nn − dx < ∞ . This conclusion gives more precise information than (1.1) when u k → u = 0 weakly in W ,n (Ω). Based on the result of Lions and the blowing up analysis method, Adimurthi sharp Trudinger-Moser type inequality in R n R on boundeddomains Ω, which can be described as followssup u ∈ W , (Ω) k∇ u k ≤ ˆ R e π | u | ( α k u k ) dx < ∞ , iff α < inf u ∈ W , (Ω) ,u =0 k∇ u k k u k . Subsequently, this result was extended to L p norm in two dimension and high dimension aswell in Yang [29], Lu and Yang [20], [21] and Zhu [30].Another interesting extension of (1.1) is to construct Trudinger-Moser inequalities forunbounded domains. In fact, we note that, even in the case α < α n , the supremum in (1.1)becomes infinite for domains Ω ⊆ R n with | Ω | = + ∞ . Related inequalities for unboundeddomains have been first considered by D.M. Cao [5] in the case N = 2 and for any dimensionby J.M. do ´O [10] and Adachi-Tanaka [1] in the subcritical case, that is α < α n . In [24],B. Ruf showed that in the case N = 2, the exponent α = 4 π becomes admissible if theDirichlet norm ´ Ω |∇ u | dx is replaced by Sobolev norm ´ Ω (cid:0) | u | + |∇ u | (cid:1) dx , more precisely,he proved that(1.2) sup u ∈ W , ( R ) ´ R ( | u | + |∇ u | ) dx ≤ ˆ R Φ (cid:0) α | u | (cid:1) dx < + ∞ , iff α ≤ π, where Φ ( t ) = e t −
1. Later, Y. X. Li and B. Ruf [17] extended Ruf’s result to arbitrarydimension.Recently, M. de Souza and J. M. do ´O [9] obtained an Adimurthi-Druet type result in R for some weighted Sobolev space E = (cid:26) u ∈ W , (cid:0) R (cid:1) : ˆ R V ( x ) u dx < ∞ (cid:27) , where the potential V is radially symmetric, increasing and coercive.In this paper, we will try to remove the potential V in [9], and we obtain an Adimurthi-Druet type result for W ,n ( R n ). Our main results read as follows Theorem 1.1.
For any ≤ α < , the following holds: (1.3) sup k u k W ,n ( R n ) =1 ˆ R n Φ (cid:16) α n | u | nn − (1 + α k u k nn ) n − (cid:17) dx < ∞ , where Φ ( t ) = e t − n − P j =0 t j j ! . Moreover, for any α ≥ , the supremum is infinite. G. Lu and M. Zhu
At this point, we call attention to the recent work of M. de Souza and J. M. do ´O in [12],where the authors establish an analogue of (1.3) under the additional assumption that Φ ( t )is substituted by a smaller function Ψ ( t ) = e t − n − P j =0 t j j ! . But, they did not address whether thesupremum is finite when α = 1. Here, we remark that by using the test function sequenceconstructed in Section 2, we can show that the supreme in (1.3) is infinity when α = 1.Therefore, our results indeed improve substantially the result in [12].We set S = sup k u k W ,n ( R n ) =1 ˆ R n Φ (cid:16) α n | u | nn − (1 + α k u k nn ) n − (cid:17) dx. The existence of an extremal function for the above supremum is only known when α = 0as shown in [17]. However, whether an extremal function for the above supremum exists ornot is not known for α >
0. Our next aim is to show that the supremum above is attainedwhen α is chosen small enough, that is Theorem 1.2.
There exists u α ∈ W ,n ( R n ) with k u α k W ,n ( R n ) = 1 such that S = ˆ R n Φ (cid:16) α n | u α | nn − (1 + α k u α k nn ) n − (cid:17) dx for sufficiently small α . The first result about existence of the extremal function for Trudinger-Moser inequalitywas given by L. Carleson and S.Y.A. Chang in [6], where it is proved that the supremumin (1.1) indeed has extremals by using symmetrization argument, when Ω is a ball in R n .This actually brings a surprise, since it is well-known that the Sobolev-inequality has noextremals on any finite domain Ω = R n . Later, M. Flucher [13] showed that this resultcontinues to hold for any smooth domain in R and Lin in [18] generalized the result toany dimension. More existence results can be found in several papers, see e.g. Y.X. Li [15]and [16] for Trudinger-Moser inequalities on compact Riemannian manifold, [24] and [17]for on unbounded domains in R n , and Lu and Yang [20],[21] and Zhu [30] for Trudinger-Moser inequalities involving a remainder term. For the existence of critical points for thesupercritical regime, i.e. the Trudinger-Moser energy functionals constrained to manifold M = n u ∈ W ,n (Ω) , k∇ u k L n ((Ω)) > o , see del Pino, Musso and Ruf [8] and Malchiodi andMartinazzi [22], and references therein.We now sketch the idea of proving Theorem 1.1 and Theorem 1.2.1. The proof of the second part of Theorem 1.1 is based on a test function argument.Unlike in the case for bounded domains [29], we cannot construct the test function by theeigenfunction of the first eigenvalue problem:inf u ∈ W ,n (Ω) ,u =0 k∇ u k nn k u k nn , sharp Trudinger-Moser type inequality in R n R n . To overcome this difficulty,we will construct a new test function sequence (see Section 2 for more details).2. For the proof of the first part of Theorem 1.1, we will carry out the standard blowingup analysis procedure. This method is based on a blowing up analysis of sequences ofsolutions to n -Laplacian in R n with exponential growth, and it has been successfully appliedin the proof of the Trudinger-Moser inequalities and related existence results in boundeddomains (see [4],[30],[20] and [21]). In the unbounded case, one will encounter many newdifficulties. For instance, when the blowing up phenomenon arises, a crucial step is to showthe strong convergence of u k in L n norm ( u k are the maximizers for a sequence of subcriticalTrudinger-Moser energy functionals). We recall that in [12], the authors proved the strongconvergence under the additional assumption that Φ ( t ) is substituted by a smaller functionΨ ( t ) = e t − n − P j =0 t j j ! . In our case, we remove that unnatural assumption, and in order to provethe strong convergence, we will need more careful analysis and different technique (see § . c denotes a constant which may vary from line to line. In this section, we prove the sharpness of the inequality in Theorem 1.1. Namely, we willshow that if α ≥
1, then the sumpremum is infinity.
Proof of the Second Part of Theorem 1.1.
Setting u k = 1 ω n n − ( (log k ) − n log R k | x | R k k < | x | ≤ R k (log k ) n − n < | x | ≤ R k k G. Lu and M. Zhu where R k := (log k ) / n log log k → ∞ , as k → ∞ . We can easily verify that ˆ R n |∇ u k | n dx = 1and k u k k nn = ˆ B Rk/k + ˆ B Rk \ B Rk/k ! | u k | n dx = (log k ) n − n (cid:18) R k k (cid:19) n + R nk log k ˆ k (log r ) n r n − dr = C n R nk log k (1 + o (1)) → k → ∞ , where C n = ´ k (log r ) n r n − dr . Therefore we have k u k k nW ,n ( R n ) = 1 + C n R nk log k (1 + o (1))Since 1 + k u k k nn k u k k nW ,n = 1 + 2 k u k k nn k u k k nn , then on the ball B R k /k , we have α n | u k | nn − k u k k nn − W ,n ( R n ) k u k k nn k u k k nW ,n ( R n ) ! n − = nω n − n − | u k | nn − (1 + 2 k u k k nn ) n − (1 + k u k k nn ) n − = n log k (cid:18) − n − k u k k nn + 2 n − k u k k nn (1 + o (1)) (cid:19) and (cid:12)(cid:12) B R k /k (cid:12)(cid:12) = ω n − n exp ( n log R k − n log k ).Thussup k u k W ,n ( R n ) =1 ˆ R n Φ (cid:16) α n | u | nn − (1 + k u k nn ) (cid:17) dx ≥ c ˆ B Rk/k exp α n | u k | nn − k u k k nn − W ,n (cid:18) k u k k nn k u k k nW ,n (cid:19)! dx sharp Trudinger-Moser type inequality in R n ≥ c exp (cid:18) n log k (cid:18) − n − k u k k nn + 2 n − k u k k nn (1 + o (1)) (cid:19) + n log R k − n log k (cid:19) = c exp (cid:18) n log k (cid:18) − n − k u k k nn + 2 n − k u k k nn (1 + o (1)) (cid:19) + n log R k (cid:19) Because n log R k = n log (log k ) / n log log k ! = 12 log log k − n log log log k and n log k (cid:18) − n − k u k k nn + 2 n − k u k k nn (1 + o (1)) (cid:19) = − nn − C n R nk log k (1 + o (1))= − nn − C n k ) n (1 + o (1)) , we can get ˆ R n Φ (cid:16) α n | u k | nn − (1 + k u k k nn ) (cid:17) dx ≥ c exp (cid:0) n log k (cid:0) − k u k k nn (1 + o (1)) (cid:1) + n log R k (cid:1) = c exp (cid:18)
12 log log k − n log log log k − nC n n − k ) n (1 + o (1)) (cid:19) → ∞ . The proof is finished.
We first present a technical lemma contributed by Jo˜ao Marco do ´O, et al [11].
Lemma 3.1.
Let { u k } be a sequence in W ,n ( R n ) such that k u k k W ,n = 1 and u k → u = 0 ,weakly in W ,n ( R n ) . If < q < q n ( u ) := 1(1 − k u k nW ,n ) / ( n − , then sup k ˆ R n Φ (cid:16) α n q | u k | nn − (cid:17) dx < ∞ . G. Lu and M. Zhu
Let { R k } be an increasing sequence which diverges to infinity, and { β k } an increasingsequence which converges to α n . Setting I αβ k ( u ) = ˆ B Rk Φ (cid:16) β k | u | nn − (1 + α k u k nn ) n − (cid:17) dx and H = (cid:8) u ∈ W ,n ( B R k ) (cid:12)(cid:12) k u k W ,n = 1 (cid:9) . We have
Lemma 3.2.
For any ≤ α ≤ , there exists an extremal function u k ∈ H such that I αβ k ( u k ) = sup u ∈ H I αβ k ( u ) Proof.
There exists a sequence of { v i } ∈ H such thatlim i →∞ I αβ k ( v i ) = sup u ∈ H I αβ k ( u ) . Since v i is bounded in W ,n ( R n ), there exists a subsequence which will still be denoted by v i , such that v i → u k weakly in W ,n ( R n ) ,v i → u k strongly in L s ( B R k ) , for any 1 < s < ∞ as i → ∞ . Hence v i → u k a.e. in R n , and g i = Φ n β k | v i | nn − (1 + α k v i k nn ) n − o → g k = Φ n β k | u k | nn − (1 + α k u k k nn ) n − o a.e. in R n . We claim that u k = 0. If not, 1 + α k v i k nn →
1, and then g i is bounded in L r ( B R k )for some r >
1, thus g i →
0. Therefore, sup u ∈ H I αβ k ( u ) = 0, which is impossible. By Lemma3.1, we have for any q < q n ( u k ) := ( −k u k k W ,n ) / ( n − ,lim sup i →∞ ˆ R n Φ (cid:16) α n q | v i | nn − (cid:17) dx < ∞ . Since α ≤
1, we have1 + α k u k k nn < k u k k nW ,n < − k u k k nW ,n = q n ( u k ) , then g i is bounded in L s for some s >
1, and g i → g k strongly in L ( B R k ), as i → ∞ . Therefore, the extremal function is attained for the case β k < α n and k u k k W ,n = 1. sharp Trudinger-Moser type inequality in R n Lemma 3.3.
Let u k be as above. Then(i) u k is a maximizing sequence for S ; (ii) u k may be chosen to be radially symmetric and decreasing.Proof. (i) Let η be a cut-off function which is 1 on B and 0 on R n \ B . Then given any ϕ ∈ W ,n ( R n ) with ´ R n ( | ϕ | n + |∇ ϕ | n ) dx = 1, we have τ n ( L ) := ˆ R n (cid:16)(cid:12)(cid:12)(cid:12) ∇ η (cid:16) xL (cid:17) ϕ (cid:12)(cid:12)(cid:12) n + (cid:12)(cid:12)(cid:12) η (cid:16) xL (cid:17) ϕ (cid:12)(cid:12)(cid:12) n (cid:17) dx → , as L → + ∞ . Hence for a fixed L and R k > L , ˆ B L Φ β k (cid:12)(cid:12)(cid:12)(cid:12) ϕτ ( L ) (cid:12)(cid:12)(cid:12)(cid:12) nn − α (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) η (cid:0) xL (cid:1) ϕτ ( L ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) nn ! n − dx ≤ ˆ B L Φ β k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) η (cid:0) xL (cid:1) ϕτ ( L ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) nn − α (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) η (cid:0) xL (cid:1) ϕτ ( L ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) nn ! n − dx ≤ ˆ B Rk Φ (cid:16) β k | u k | nn − (1 + α k u k k nn ) n − (cid:17) dx. By the Levi Lemma, we have ˆ B L Φ α n (cid:12)(cid:12)(cid:12)(cid:12) ϕτ ( L ) (cid:12)(cid:12)(cid:12)(cid:12) nn − α (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) η (cid:0) xL (cid:1) ϕτ ( L ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) nn ! n − dx ≤ lim k →∞ ˆ R n Φ (cid:16) β k | u k | nn − (1 + α k u k k nn ) n − (cid:17) dx. Letting L → ∞ , we get ˆ R n Φ (cid:16) α n | ϕ | nn − (1 + α k ϕ k nn ) n − (cid:17) dx ≤ lim k →∞ ˆ R n Φ (cid:16) β k | u k | nn − (1 + α k u k k nn ) n − (cid:17) dx. Hence, lim k →∞ ˆ B Rk Φ (cid:16) β k | u k | nn − (1 + α k u k k nn ) n − (cid:17) dx = sup k u k W ,n ( R n ) =1 ˆ R n Φ (cid:16) α n | u | nn − (1 + α k u k nn ) n − (cid:17) dx. G. Lu and M. Zhu (ii) Let u ∗ k be the radial rearrangement of u k . Then τ nk := ˆ R n ( |∇ u ∗ k | n + | u ∗ k | n ) dx ≤ ˆ R n ( |∇ u k | n + | u k | n ) dx = 1 , thus ˆ B Rk Φ β k (cid:12)(cid:12)(cid:12)(cid:12) u ∗ k τ k (cid:12)(cid:12)(cid:12)(cid:12) nn − (cid:18) α (cid:13)(cid:13)(cid:13)(cid:13) u ∗ k τ k (cid:13)(cid:13)(cid:13)(cid:13) nn (cid:19) n − ! dx ≥ ˆ B Rk Φ (cid:16) β k | u ∗ k | nn − (1 + α k u ∗ k k nn ) n − (cid:17) dx. Since ˆ B Rk Φ (cid:16) β k | u ∗ k | nn − (1 + α k u ∗ k k nn ) n − (cid:17) dx = ˆ B Rk Φ (cid:16) β k | u k | nn − (1 + α k u k k nn ) n − (cid:17) dx, we have τ k = 1. It is well-known that τ k = 1 iff u k is radial. Therefore ˆ B Rk Φ (cid:16) β k | u ∗ k | nn − (1 + α k u ∗ k k nn ) n − (cid:17) dx = sup k u k W ,n ( BRk ) =1 ˆ B Rk exp n β k | u | nn − (1 + α k u k nn ) n − o dx. So, we can assume u k = u k ( | x | ), and u k ( r ) is decreasing. In this section, the method of blow-up analysis will be used to analyze the asymptoticbehavior of the maximizing sequence { u k } , and the first part of Theorem 1.1 will be finished.After a direct computation, the Euler-Lagrange equation for the extremal function u k ∈ W ,n ( B R k ) of I αβ k ( u ) can be written as(4.1) − △ n u k + u n − k = µ k λ − k u n − k Φ ′ n α k u nn − k o + γ k u n − k where u k ∈ W ,n ( B R k ) , k u k k W ,n = 1 ,α k = β k (1 + α k u k k nn ) n − ,µ k = (1 + α k u k k nn ) / (1 + 2 α k u k k nn ) ,γ k = α/ (1 + 2 α k u k k nn ) ,λ k = ´ B Rk u nn − k Φ ′ (cid:16) α k u nn − k (cid:17) . In the following, we denote c k = max u k = u k (0). First, we give the following importantobservation. sharp Trudinger-Moser type inequality in R n Lemma 4.1. inf k λ k > . Proof.
Assume λ k →
0. Then λ k = ˆ R n u nn − k Φ ′ (cid:16) α k u nn − k (cid:17) dx = ˆ R n u nn − k ∞ X j = n − (cid:16) α k u nn − k (cid:17) j j ! dx = ˆ R n α jk u nk ( n − . . . ! dx ≥ α jk ( n − ˆ R n u nk dx. (4.2)Since u k ( | x | ) is decreasing, we have u nk ( L ) | B L | ≤ ´ B L u nk dx ≤
1, and then(4.3) u nk ( L ) ≤ nω n − L n . Set ε = nω n − L n . Then for any x / ∈ B L , we have u k ≤ ε , and ˆ R n \ B L Φ (cid:16) α k u nn − k (cid:17) dx ≤ c ˆ R n \ B L u nk dx ≤ cλ k → . SinceΦ (cid:16) α k u nn − k (cid:17) = ∞ X j = n − (cid:16) α k u nn − k (cid:17) j j ! ≤ ∞ X j = n − α k u nn − k (cid:16) α k u nn − k (cid:17) j ( j + 1) j ! ≤ α k u nn − k Φ ′ (cid:16) α k u nn − k (cid:17) , we have lim k →∞ ˆ B L Φ (cid:16) α k u nn − k (cid:17) dx = lim k →∞ (cid:18) ˆ B L ∩{ u k ≥ } + ˆ B L ∩{ u k < } (cid:19) Φ (cid:16) α k u nn − k (cid:17) dx ≤ lim k →∞ (cid:18) c ˆ B L u nn − k Φ (cid:16) α k u nn − k (cid:17) dx + ˆ B L ∩{ u k < } Φ (cid:16) α k u nn − k (cid:17) dx (cid:19) ≤ lim k →∞ (cid:18) cλ k + c ˆ B L u nk dx (cid:19) . By (4.2), we see that ´ B L u qk dx →
0, for any q >
1, and then we havelim k →∞ ˆ B L Φ (cid:16) α k u nn − k (cid:17) dx = 0 . This is impossible.Now, we introduce the concept of Sobolev-normalized concentrating sequence and concentration-compactness principle as in [24].2
G. Lu and M. Zhu
Definition 4.1.
A sequence { u k } ∈ W ,n ( R n ) is a Sobolev-normalized concentrating se-quence, ifi) k u k k W ,n ( R n ) = 1;ii) u k → W ,n ( R n ) ;iii) there exists a point x such that for any δ > ´ R n \ B δ ( x ) ( |∇ u k | n + | u k | n ) dx → Lemma 4.2.
Let { u k } be a sequence satisfying k u k k W ,n ( R n ) = 1 , and u k → u weakly in W ,n ( R n ) . Then either { u k } is a Sobolev-normalized concentrating sequence, or there exists γ > such that Φ (cid:16) ( α n + γ ) | u k | nn − (cid:17) is bounded in L ( R n ) . Lemma 4.3. If sup k c k < ∞ , then Theorem 1.1 and Theorem 1.2 hold.Proof. For any ε >
0, by using (4.3) we can find some L such that u k ( x ) ≤ ε when x / ∈ B L .We rewrite ´ R n (cid:16) Φ (cid:16) α k u nn − k (cid:17) − α n − k u nk ( n − (cid:17) dx as (cid:18) ˆ B L + ˆ R n \ B L (cid:19) (cid:18) Φ (cid:16) α k u nn − k (cid:17) − α n − k u nk ( n − (cid:19) dx. Since ˆ R n \ B L (cid:18) Φ (cid:16) α k u nn − k (cid:17) − α n − k u nk ( n − (cid:19) dx = c ˆ R n \ B L u n n − k dx ≤ cε n n − − n ˆ R n u nk dx = cε n n − − n , we have(4.4) ˆ R n (cid:18) Φ (cid:16) α k u nn − k (cid:17) − α n − k u nk ( n − (cid:19) dx = ˆ B L (cid:18) Φ (cid:16) α k u nn − k (cid:17) − α n − k u nk ( n − (cid:19) dx + O (cid:16) ε n n − − n (cid:17) . It follows from sup k c k < ∞ that ˆ R n Φ (cid:16) α k u nn − k (cid:17) dx = ˆ B L (cid:18) Φ (cid:16) α k u nn − k (cid:17) − α n − k u nk ( n − (cid:19) dx + ˆ R n α n − k u nk ( n − dx + O (cid:16) ε n n − − n (cid:17) ≤ c ( L ) , thus, Theorem 1.1 holds. By Lemma 4.1 and applying the elliptic estimate in [27] to equation(4.1), we have u k → u in C loc ( R n ).When u = 0, we claim that { u k } is not a Sobolev-normalized concentrating sequence. Ifnot, by iii) of Definition 4.1 and the fact that | u k | is bounded, we have for any δ > ˆ R n u nk dx ≤ ˆ B δ u nk dx + ˆ R n \ B δ u nk dx sharp Trudinger-Moser type inequality in R n ≤ cδ n + o k (1) . Letting δ →
0, we have ´ R n u nk dx →
0, as k → ∞ . For any ε >
0, when L is large enough,we have by (4.4) that S + o k (1) = ˆ R n Φ (cid:16) α k u nn − k (cid:17) dx = ˆ R n α n − k u nk ( n − dx + ˆ B L (cid:18) Φ (cid:16) α k · u nn − (cid:17) − ˆ B L α n − k · u n ( n − (cid:19) dx + O (cid:16) ε n n − − n (cid:17) , then S ≤ ˆ R n α n − k u nk ( n − dx → , which is impossible, and thus the claim is proved.By Lemma 4.2, we have ´ R n Φ (cid:16) α k u nn − k (cid:17) dx → ´ R n Φ (cid:16) α n u nn − (cid:17) dx = 0, which is stillimpossible. Therefore, u = 0.Now, we show that ´ R n u nk → ´ R n u n . By (4.4), we have S = lim k →∞ ˆ R n Φ (cid:16) α k u nn − k (cid:17) dx = ˆ R n (cid:16) Φ (cid:16) lim k →∞ α k u nn − (cid:17)(cid:17) dx + lim k →∞ ˆ R n lim k →∞ α n − k ( u nk − u n )( n − dx. (4.5)Set τ nk = ´ R n u nk ´ R n u n . By the Levi Lemma, we have τ k ≥
1. Let ˜ u = u (cid:16) xτ k (cid:17) . Then, we have ˆ R n |∇ ˜ u | n dx = ˆ R n |∇ u | n dx ≤ ˆ R n |∇ u k | n dx and ˆ R n | ˜ u | n dx = τ nk ˆ R n | u | n dx ≤ ˆ R n | u k | n dx. Therefore ˆ R n ( |∇ ˜ u | n + | ˜ u | n ) dx ≤ . Hence, we have by (4.5) that S ≥ ˆ R n Φ (cid:16) α n (1 + α k ˜ u k nn ) n − ˜ u nn − (cid:17) dx G. Lu and M. Zhu = τ n ˆ R n Φ (cid:16) α n (1 + ατ n k u k nn ) n − u nn − (cid:17) dx ≥ τ n ˆ R n Φ (cid:16) lim k →∞ α k u nn − (cid:17) dx + o (1)= ˆ R n Φ (cid:16) lim k →∞ α k u nn − (cid:17) + ( τ n − ˆ R n lim k →∞ α n − k u n ( n − dx ++ ( τ n − ˆ R n Φ (cid:16) lim k →∞ α k u nn − (cid:17) − ˆ R n lim k →∞ α n − k u n ( n − + o (1)= ( τ n − ˆ R n Φ (cid:16) lim k →∞ α k u nn − (cid:17) − ˆ R n lim k →∞ α n − k u n ( n − ++ lim k →∞ ˆ R n Φ (cid:16) α k u nn − k (cid:17) dx + o (1)= S + ( τ n − ˆ R n Φ (cid:16) lim k →∞ α k u nn − (cid:17) − ˆ R n lim k →∞ α n − k u n ( n − dx + o (1)Since Φ (cid:16) lim k →∞ α k u nn − (cid:17) − ´ R n lim k →∞ α n − k u n ( n − >
0, we have τ = 1, thenlim k ˆ R n Φ (cid:16) α k u nn − k (cid:17) dx = ˆ R n Φ (cid:16) α n (1 + α k u k nn ) n − u nn − (cid:17) dx Thus, u is an extremal function.In the following, we assume c k → + ∞ and perform a blow-up procedure. u k In this subsection, we investigate the asymptotic behavior of u k . First, we introduce thefollowing important quatity r nk = λ k µ k c nn − k e α k c nn − k . By (4.3), we can find a sufficiently large L such that u k ≤ R n \ B L . Then ( u k − u k ( L )) + ∈ W ,n ( B L ) and sharp Trudinger-Moser type inequality in R n ˆ B L (cid:12)(cid:12) ∇ ( u k − u k ( L )) + (cid:12)(cid:12) n dx ≤ , hence by [29, Theorem 1.1], we have ˆ B L e α n ( β k u k − u k ( L ) k nn ) n − ( u k − u k ( L )) nn − dx ≤ c ( L ) , provided β < inf u ∈ W ,n ( B L ) k∇ u k nn k u k nn . For any q < α n (1 + β k u k − u k ( L ) k nn ) n − , we can find a constant c ( q ) such that qu nn − k ≤ α n (1 + β k u k − u k ( L ) k nn ) n − (cid:0) ( u k − u k ( L )) + (cid:1) nn − + c ( q ) , and then we have(4.6) ˆ B L e qu nn − k dx ≤ c ( L, q ) . Now we take some 0 < A < − A ) β k (1 + α k u k k nn ) n − < α n (1 + β k u k − u k ( L ) k nn ) n − . Then λ k e − Aβ k ( α k u k k nn ) n − c nn − k = e − Aβ k ( α k u k k nn ) n − c nn − k (cid:20)(cid:18) ˆ R n \ B L + ˆ B L (cid:19) u nn − k Φ ′ (cid:16) α k u nn − k (cid:17) dx (cid:21) ≤ ce − Aβ k ( α k u k k nn ) n − c nn − k (cid:18) ˆ R n \ B L u nk dx + ˆ B L u nn − k e β k ( α k u k k nn ) n − u nn − k dx (cid:19) ≤ c ˆ B L u nn − k e (1 − A ) β k ( α k u k k nn ) n − u nn − k dx + o (1) . Since u k converges strongly in L s ( B L ) for any s >
1, by using H¨older’s inequality and (4.6),we have λ k ≤ ce Aα k c nn − k , hence(4.7) r nk ≤ Ce ( A − α k c nn − k = o (cid:0) c − qk (cid:1) , G. Lu and M. Zhu for any q > m k ( x ) = u k ( r k x ) ,φ k ( x ) = m k ( x ) c k ,ψ k ( x ) = nn − α k c n − k ( m k − c k ) , where m k , φ k and ψ k are defined on Ω k := { x ∈ R n : r k x ∈ B } . From (4.1) and (4.7), weknow φ k ( x ) , ψ k ( x ) satisfy −△ n φ k ( x ) = r nk c n − k (cid:18) µ k λ − k m n − k Φ ′ n α k m nn − k o + ( γ k − m n − k (cid:19) = (cid:18) c nk φ n − k ( x ) Φ ′ n α k (cid:16) m nn − k − c nn − k (cid:17)o + o (1) (cid:19) (4.8)and −△ n ψ k ( x ) = (cid:18) nα k n − (cid:19) n − c k r nk (cid:18) µ k λ − k m n − k Φ ′ n α k m nn − k o + ( γ k − m n − k (cid:19) = (cid:18) nα k n − (cid:19) n − (cid:18) m k c k (cid:19) n − e α k (cid:18) m nn − k − c nn − k (cid:19) + o (1) ! . (4.9)We analyze the limit function of φ k and ψ k ( x ). Since u k is bounded in W ,n ( R n ), thereexists a subsequence such that u k → u weakly in W ,n ( R n ). Because the right side of (4.8)vanishes as k → ∞ , then we have φ k → φ in C loc ( R n ), as k → ∞ , by applying the classicaleatimates [27]. Therefore, −△ n φ = 0 in R n . Since φ k (0) = 1, by the Lionville-type theorem, we have φ ≡ R n .Now, we investigate the asymptotic behavior of ψ k . By (4.7) and the fact that φ k ( x ) ≤ −△ n ψ k ( x ) = O (1) . By [25, Theorem 7], we know that osc B L ψ k ≤ c ( L ) for any L > . Then from the resultof [27], we have k ψ k k C ,δ ( B L ) ≤ c ( L ) for some δ >
0. Hence ψ k converges in C loc ( B L ) and m k − c k → C loc ( B L ).Since m nn − k = c nn − k (cid:18) m k − c k c k (cid:19) nn − = c nn − k (cid:18) nn − m k − c k c k + O (cid:18) c k (cid:19)(cid:19) , sharp Trudinger-Moser type inequality in R n α k (cid:16) m nn − k − c nn − k (cid:17) = α k c nn − k (cid:18) nn − m k − c k c k + O (cid:18) c k (cid:19)(cid:19) (4.10) = ψ k ( x ) + o (1) → ψ ( x ) in C loc , and then(4.11) − △ n ψ = (cid:18) nc n n − (cid:19) n − exp { ψ ( x ) } , where c n = lim k →∞ α k = α n (cid:16) α lim k →∞ k u k k nn (cid:17) n − .Since ψ is radially symmetric and decreasing, it is easy to see that (4.11) has only onesolution. We can check that ψ ( x ) = − n log (cid:18) c n n nn − | x | nn − (cid:19) and ˆ R n e ψ ( x ) dx = ω n − n − n n nn − c n ! n − ˆ ∞ (1 + t ) − n t n − dt = ω n − n − n n nn − c n ! n − · n − α lim k →∞ k u k k nn . (4.12)For any A >
1, let u Ak = min (cid:8) u k , c k A (cid:9) . Lemma 4.4.
For any
A > , there holds lim sup k →∞ ˆ R n (cid:0)(cid:12)(cid:12) u Ak (cid:12)(cid:12) n + (cid:12)(cid:12) ∇ u Ak (cid:12)(cid:12) n (cid:1) dx ≤ − A − A
11 + α lim k →∞ k u k k nn Proof.
Since (cid:12)(cid:12)(cid:8) x : u k ≥ c k A (cid:9)(cid:12)(cid:12) (cid:12)(cid:12) c k A (cid:12)(cid:12) n ≤ ´ { u k ≥ ckA } | u k | n dx ≤
1, we can find a sequence ρ k → n x : u k ≥ c k A o ⊂ B ρ k . Since u k converges in L s ( B ) for any s >
1, we havelim k →∞ ˆ { u k ≥ ckA } (cid:12)(cid:12) u Ak (cid:12)(cid:12) s dx ≤ lim k →∞ ˆ { u k ≥ ckA } | u k | s dx = 0 , G. Lu and M. Zhu and then for any s >
0, lim k →∞ ˆ R n (cid:16) u k − c k A (cid:17) + | u k | s dx = 0 . Testing (4.1) with (cid:0) u k − c k A (cid:1) + we have ˆ R n (cid:18)(cid:12)(cid:12)(cid:12)(cid:12) ∇ (cid:16) u k − c k A (cid:17) + (cid:12)(cid:12)(cid:12)(cid:12) n + (cid:16) u k − c k A (cid:17) + | u k | n − (cid:19) dx = ˆ R n (cid:16) u k − c k A (cid:17) + µ k λ − k u n − k Φ ′ n α k u nn − k o dx + o (1) ≥ ˆ B Rrk (cid:16) u k − c k A (cid:17) + µ k λ − k u n − k exp n α k u nn − k o dx + o (1)= ˆ B R (cid:0) m k − c k A (cid:1) c k (cid:18) m k − c k c k + 1 (cid:19) n − exp { ψ k ( x ) + o (1) } dx + o (1) ≥ A − A ˆ B R e ψ ( x ) dx. Letting R → ∞ , k → ∞ , by (4.12), we havelim inf k →∞ ˆ R n (cid:18)(cid:12)(cid:12)(cid:12)(cid:12) ∇ (cid:16) u k − c k A (cid:17) + (cid:12)(cid:12)(cid:12)(cid:12) n + (cid:16) u k − c k A (cid:17) + | u k | n − (cid:19) dx ≥ A − A (cid:16) α lim k →∞ k u k k nn (cid:17) . Now, observe that ˆ R n (cid:0)(cid:12)(cid:12) ∇ u Ak (cid:12)(cid:12) n + (cid:12)(cid:12) u Ak (cid:12)(cid:12) n (cid:1) dx = 1 − ˆ R n (cid:18)(cid:12)(cid:12)(cid:12)(cid:12) ∇ (cid:16) u k − c k A (cid:17) + (cid:12)(cid:12)(cid:12)(cid:12) n + (cid:16) u k − c k A (cid:17) + | u k | n − (cid:19) dx + ˆ R n (cid:16) u k − c k A (cid:17) + | u k | n − dx − ˆ { u k > ckA } | u k | n dx + ˆ { u k > ckA } (cid:12)(cid:12) u Ak (cid:12)(cid:12) n dx ≤ − A − A
11 + α lim k →∞ k u k k nn + o (1) , the proof is finished. Lemma 4.5. lim k k u k k nn = 0 .Proof. If { u k } is a Sobolev-normalized concentrating sequence, then lim k k u k k nn = 0. If { u k } is not a Sobolev-normalized concentrating sequence, and lim k k u k k nn = 0. For A large enough,there exist some constant ε > ˆ R n (cid:0)(cid:12)(cid:12) ∇ u Ak (cid:12)(cid:12) n + (cid:12)(cid:12) u Ak (cid:12)(cid:12) n (cid:1) dx = 1 −
11 + ( α + ε ) lim k k u k k nn < . sharp Trudinger-Moser type inequality in R n ´ R n Φ (cid:16) qα n (cid:12)(cid:12) u Ak (cid:12)(cid:12) nn − (cid:17) dx ≤ ∞ , for any q < (cid:16) α + ε ) lim k k u k k nn (cid:17) ( α + ε ) lim k k u k k nn n − . Since α < k u k k W ,n = 1 and lim k k u k k nn = 0, we can take some ε such that ( α + ε ) lim k k u k k nn <
1, and then (cid:16) α lim k k u k k nn (cid:17) n − < α + ε ) lim k k u k k nn ( α + ε ) lim k k u k k nn n − . Therefore(4.13) ˆ R n Φ (cid:16) p ′ α k (cid:12)(cid:12) u Ak (cid:12)(cid:12) nn − (cid:17) dx < ∞ , for some p ′ > n u k ∈ L r for some r > ´ { u k > ckA } |∇ u k | n dx →
0. In this case, we can easily derive the above claim by theTrudinger-Moser inequalities on bounded domains and (4.13). When ´ { u k > ckA } |∇ u k | n dx ≥ c ,for some c >
0. We split u k as u k + u k , with u k → cδ and ´ { u k > ckA } |∇ u k | n dx →
0. Since α <
1, we have1 + α k u k k nn = 1 + α (cid:13)(cid:13) u k (cid:13)(cid:13) nn + o k (1) < − k u k k nW ,n + o k (1) ≤ k∇ u k k nn + o k (1) ≤ k∇ u k k nL n ( { u k > ckA } ) + o k (1) , and then there exists some constant s > α k u k k nn ) s ≤ k∇ u k k nLn ( { uk> ckA } ) ,therefore by the classic Trudinger-Moser inequality on the bounded domain and (4.13), theclaim is proved.Based on the the claim above and the classic elliptic estimate, we know that u k is boundednear 0, and which contradicts the assumption that c k → ∞ . Therefore lim k k u k k nn = 0, andthe lemma is proved. Remark 4.1.
From the above lemma, we havelim k α k = α n , lim k µ k = 1 , G. Lu and M. Zhu lim sup k →∞ ˆ R n (cid:0)(cid:12)(cid:12) ∇ u Ak (cid:12)(cid:12) n + (cid:12)(cid:12) u Ak (cid:12)(cid:12) n (cid:1) dx = 1 A ,ψ ( x ) = − n log (cid:18) (cid:16) ω n − n (cid:17) n − | x | nn − (cid:19) , and lim R →∞ lim k →∞ λ k ˆ B Rrk u nn − k exp (cid:16) α k u nn − k (cid:17) dx = lim R →∞ lim k →∞ µ k ˆ B R e ψ ( x ) dx = lim k →∞ (cid:16) α lim k k u k k nn (cid:17) µ k = 1 . (4.14) Corollary 4.1.
We have lim k →∞ ´ R n \ B δ ( |∇ u k | n + | u k | n ) dx = 0 , for any δ > , and then lim k →∞ u k : ≡ . Lemma 4.6.
We have (4.15) lim k →∞ ˆ R n Φ (cid:16) α k u nn − k (cid:17) dx ≤ lim R →∞ lim k →∞ ˆ B Rrk (cid:16) exp (cid:16) α k u nn − k (cid:17) − (cid:17) dx = lim sup k →∞ λ k c nn − k , moreover, (4.16) λ k c k → ∞ and sup k c nn − k λ k ≤ ∞ . .Proof. For any
A >
1, from the expression of λ k , we have ˆ R n Φ (cid:16) α k u nn − k (cid:17) dx ≤ ˆ u k < ckA Φ (cid:16) α k u nn − k (cid:17) dx + ˆ u k ≥ ckA Φ ′ (cid:16) α k u nn − k (cid:17) dx ≤ ˆ R n Φ (cid:16) α k (cid:0) u Ak (cid:1) nn − (cid:17) dx + ˆ u k ≥ ckA Φ ′ (cid:16) α k u nn − k (cid:17) dx ≤ ˆ R n Φ (cid:16) α k (cid:0) u Ak (cid:1) nn − (cid:17) dx + (cid:18) Ac k (cid:19) nn − λ k ˆ u k ≥ ckA u nn − k λ k Φ ′ (cid:16) α k u nn − k (cid:17) dx. Thanks to Remark 4.1 and [17, Theorem 1.1], Φ (cid:16) α k (cid:0) u Ak (cid:1) nn − (cid:17) is bounded in L r for some r >
1. Since u Ak → R n as k → ∞ , we have ˆ R n Φ (cid:16) α k (cid:0) u Ak (cid:1) nn − (cid:17) dx → ˆ R n Φ (0) dx = 0, as k → ∞ . sharp Trudinger-Moser type inequality in R n k →∞ ˆ R n Φ (cid:16) α k u nn − k (cid:17) dx ≤ (cid:18) Ac k (cid:19) nn − λ k ˆ u k ≥ ckA u nn − k λ k Φ ′ (cid:16) α k u nn − k (cid:17) dx + o (1)= lim k →∞ A nn − λ k c nn − k + o (1)Letting A → k → ∞ we obtain (4.15).If λ k c k is bounded or sup k c nn − k λ k = ∞ , from (4.15), we havelim k →∞ ˆ R n Φ (cid:16) α k u nn − k (cid:17) dx = 0 , which is impossible. Lemma 4.7.
For any ϕ ∈ C ∞ ( R n ) , we have ˆ R n ϕµ k λ − k c k u n − k Φ ′ (cid:16) α k u nn − k (cid:17) dx = ϕ (0) . Proof.
As [17, Lemma 3.6], we split the integral as follows ˆ R n ϕµ k λ − k c k u n − k Φ ′ (cid:16) α k ( u k ) nn − (cid:17) dx ≤ ˆ { u k ≥ ckA } \ B Rrk + ˆ B Rrk + ˆ { u k < ckA } ! . . . dx = I + I + I . Now, we have I ≤ A k ϕ k L ∞ ˆ { u k ≥ ckA } \ B Rrk µ k λ − k c k u n − k Φ ′ (cid:16) α k ( u k ) nn − (cid:17) dx ≤ A k ϕ k L ∞ ˆ R n − ˆ B Rrk ! µ k λ − k u nn − k Φ ′ (cid:16) α k ( u k ) nn − (cid:17) dx ≤ A k ϕ k L ∞ (cid:18) − ˆ B R exp (cid:16) α k m nn − k − α k c nn − k (cid:17)(cid:19) = A k ϕ k L ∞ (cid:18) − ˆ B R exp ( ψ k ( x ) + o (1)) (cid:19) G. Lu and M. Zhu and I = ˆ B Rrk ϕµ k λ − k c k u n − k Φ ′ (cid:16) α k u nn − k (cid:17) dx = ˆ B R ϕ ( r k x ) (cid:18) m k c k (cid:19) n − exp (cid:16) α k m nn − k − α k c nn − k (cid:17) dx + o (1)= ϕ (0) ˆ B R exp ( ψ k ( x ) + o (1)) dx + o (1) = ϕ (0) + o (1) → ϕ (0) , as k → ∞ .By (4.16) and H¨older’s inequality, we obtain I = ˆ { u k < ckA } ϕµ k λ − k c k u n − k Φ ′ (cid:16) α k ( u k ) nn − (cid:17) dx = ˆ R n ϕµ k λ − k c k (cid:0) u Ak (cid:1) n − Φ ′ (cid:16) α k (cid:0) u Ak (cid:1) nn − (cid:17) dx ≤ c k k ϕ k L ∞ λ − k (cid:18) ˆ R n (cid:0) u Ak (cid:1) qn − dx (cid:19) q (cid:18) ˆ R n Φ ′ (cid:16) q ′ α k (cid:0) u Ak (cid:1) nn − (cid:17) dx (cid:19) q ′ → , as k → ∞ , for any q ′ < A n − , such that q = q ′ q ′ − large enough. Letting R → ∞ , by Remark 4.1, thelemma is proved. Lemma 4.8.
On any Ω ⋐ R n \{ } , we have c n − k u k → G α ∈ C ,α (Ω) weakly in W ,q (Ω) forany < q < n , where G α is a Green function satisfying (4.17) − △ n G α = δ + ( α − G n − α . Proof.
Setting U k = c n − k u k . By (4.1), U k satisfy:(4.18) − △ n U k = µ k c k λ − k u n − k Φ ′ n α k u nn − k o + ( γ k − U n − k . For t ≥
1, denote U tk = min { U k , t } and Ω kt = { ≤ U k ≤ t } . Testing (4.18) with U tk , wehave ˆ R n − U tk △ n U k dx + (1 − γ k ) ˆ R n U tk U n − k dx ≤ ˆ R n U tk µ k c k λ − k u n − k Φ ′ n α k u nn − k o dx. Since γ k → α <
1, as k → ∞ , we have ˆ Ω kt (cid:12)(cid:12) ∇ U tk (cid:12)(cid:12) n dx + ˆ Ω kt (cid:12)(cid:12) U tk (cid:12)(cid:12) n dx ≤ ˆ R n (cid:0) − U tk △ n U k dx + U tk U n − k (cid:1) dx sharp Trudinger-Moser type inequality in R n ≤ c ˆ R n U tk µ k c k λ − k u n − k Φ ′ n α k u nn − k o dx ≤ ct. Let η be a radially symmetric cut-off function which is 1 on B R and 0 on B c R , and satisfy |∇ η | ≤ R large enough). Then ˆ B R (cid:12)(cid:12) ∇ ηU tk (cid:12)(cid:12) n dx ≤ ˆ B R |∇ η | n (cid:12)(cid:12) U tk (cid:12)(cid:12) n dx + ˆ B R (cid:12)(cid:12) η ∇ U tk (cid:12)(cid:12) n dx ≤ c ( R ) t + c ( R ) , taking t large enough, we have ˆ B R (cid:12)(cid:12) ∇ ηU tk (cid:12)(cid:12) n dx ≤ c ( R ) t. Then by an adaptation of an argument due to Struwe [26] (also see [17]), we can obtainthat k∇ U k k L q ( B R ) ≤ c ( q, n, α, R ) for any 1 < q < n , and thus k U k k L p ( B R ) ≤ ∞ , for any0 < p < ∞ . By Corollary 4.1, we know exp n α k u nn − k o is bounded in L r (Ω \ { B δ } ) forany r > δ >
0. Then applying [25, Theorem 2.8] and the result of [27], we have k U k k C ,α ( B R ) ≤ c , then c n − k u k → G α weakly in W ,q ( B R ). So we are done.Next, as [17, Lemma 3.8], we can obtain the following asymptotic representation of G α ,which will be used to prove the existence of Trudinger-Moser functions. Lemma 4.9. G α ∈ C ,βloc ( R n \ { } ) for some β > , and near , we have (4.19) G α = − nα n log r + A + O ( r n log n r ) . Moreover, for any δ > , we have (4.20) lim k →∞ (cid:18) ˆ R n \ B δ |∇ U k | n dx + (1 − α ) ˆ R n \ B δ U nk dx (cid:19) = ω n − | G ′ α ( δ ) | n − δ n − . Proof.
The proof of (4.19) is similar as [17, Lemma 3.8], here we only give the proof for(4.20).By Corollary 4.1, we have(4.21) ˆ R n \ B δ u nn − k Φ ′ n α k u nn − k o dx ≤ c ˆ R n \ B δ u nk dx → . Testing (4.18) with U k , we get ˆ R n \ B δ |∇ U k | n dx + ˆ ∂B δ |∇ U k | n − U k ∂U k ∂n dx = − ˆ R n \ B δ div (cid:0) |∇ U k | n − ∇ U k (cid:1) U k dx G. Lu and M. Zhu = ˆ R n \ B δ µ k c nn − k λ − k u nn − k Φ ′ n α k u nn − k o dx + ˆ R n \ B δ ( γ k − U n − k U k dx. By (4.21), (4.16), we havelim k →∞ ˆ R n \ B δ |∇ U k | n dx = − lim k →∞ ˆ ∂B δ |∇ U k | n − U k ∂U k ∂n dx + ( α −
1) lim k →∞ ˆ R n \ B δ U n − k U k dx = − G α ( δ ) ˆ ∂B δ |∇ G α | n − ∂G α ∂n dx + ( α −
1) lim k →∞ ˆ R n \ B δ U n − k U k dx = ω n − | G ′ α ( δ ) | n − δ n − + ( α −
1) lim k →∞ ˆ R n \ B δ U n − k U k dx Thus lim k →∞ (cid:18) ˆ R n \ B δ |∇ U k | n dx + (1 − α ) ˆ R n \ B δ U nk dx (cid:19) = ω n − | G ′ α ( δ ) | n − δ n − . The proof is finished.
Proof for the first part of Theorem 1.1.
By (4.3), we can choose some
L > u k ( L ) <
1, and then ˆ R n \ B L exp n β k | u k | nn − (1 + α k u k k nn ) n − o dx ≤ c ˆ R n \ B L | u k | n dx ≤ c. Now, we consider the case on B L . Since ( u k − u k ( L )) + ∈ W ,n ( B L ) and for some c > u nn − k = (cid:0) ( u k − u k ( L )) + + u k ( L ) (cid:1) nn − ≤ (cid:0) ( u k − u k ( L )) + (cid:1) nn − + c (cid:0) ( u k − u k ( L )) + (cid:1) n − u k ( L ) + u k ( L ) nn − , by Lemma 4.8, we know c n − k u k → G α , then u k ( L ) = G α ( L ) c n − k . Therefore, we have u nn − k ≤ (cid:0) ( u k − u k ( L )) + (cid:1) nn − + c (cid:18) ( u k − u k ( L )) + c k (cid:19) n − + u k ( L ) nn − ≤ (cid:0) ( u k − u k ( L )) + (cid:1) nn − + c. Thus ˆ B L exp n β k | u k | nn − (1 + α k u k k nn ) n − o dx sharp Trudinger-Moser type inequality in R n ≤ c ˆ B L exp n β k (cid:0) ( u k − u k ( L )) + (cid:1) nn − (1 + α k u k k nn ) n − o dx ≤ c ˆ B L exp n β k (cid:0) ( u k − u k ( L )) + (cid:1) nn − (cid:16) (1 + α k u k k nn ) n − − (cid:17)o exp (cid:16) β k (cid:0) ( u k − u k ( L )) + (cid:1) nn − (cid:17) dx ≤ c exp n β k c nn − k (cid:16) (1 + α k u k k nn ) n − − (cid:17)o ˆ B L exp (cid:16) β k (cid:0) ( u k − u k ( L )) + (cid:1) nn − (cid:17) dx. From Lemma 4.8 and Lemma 4.9, we know (cid:13)(cid:13)(cid:13)(cid:13) c n − k u k (cid:13)(cid:13)(cid:13)(cid:13) n is bounded. Recalling the fact that k u k k nn →
0, and applying the classic Trudinger-Moser inequality, we have ˆ B L exp n β k | u k | nn − (1 + α k u k k nn ) n − o dx ≤ c exp ( αβ k c nn − k n − k u k k nn ) ˆ B L exp (cid:16) β k (cid:0) ( u k − u k ( L )) + (cid:1) nn − (cid:17) dx = c exp (cid:26) β k αn − (cid:13)(cid:13)(cid:13)(cid:13) c n − k u k (cid:13)(cid:13)(cid:13)(cid:13) nn (cid:27) ˆ B L exp (cid:16) β k (cid:0) ( u k − u k ( L )) + (cid:1) nn − (cid:17) dx ≤ c. In this section, we will show that the existence of the extremal functions of the Trudinger-Moser ineuqality involving L n norm in R n . For this, we first establish the upper bound forcritical functional when c k → ∞ , and then construct an explicit test function, which providesa lower bound for the supremum of our Trudinger-Moser inequality, meanwhile, this lowerbound equals to the upper bound.In order to prove the existence of the extremal functions, we need the following famousresult due to L. Carleson and S.Y.A. Chang [6], which often plays a key role in proof ofexistence result (see [17], [29], [21] and [30]). Theorem 5.1 (Carleson and Chang) . Let B be a unit ball in R n . Given a function sequence { u k } ⊂ W ,n ( B ) with k∇ u k k n = 1 . If u k → weakly in W ,n ( B ) , then lim sup k →∞ ˆ B e α n | u k | nn − dx ≤ B (cid:16) e + ... + n − (cid:17) . G. Lu and M. Zhu
Proposition 5.1. If S can not be attained, then S ≤ ω n − n exp (cid:26) α n A + 1 + 12 + . . . + 1 n − (cid:27) , where A is the constant in (4.19).Proof. By the Lemma 4.9, we get ˆ R n \ B δ ( |∇ u k | n + | u k | n ) dx = c − nn − k (cid:18) α ˆ R n \ B δ U nk dx + G α ( δ ) ω n − | G ′ ( δ ) | n − δ n − + o k (1) (cid:19) = c − nn − k (cid:18) α lim k →∞ k U k k nn − nα n log δ + A + o k (1) + O δ (1) (cid:19) . Setting ¯ u k ( x ) = ( u k ( x ) − u k ( δ )) + . Then we have ˆ B δ |∇ ¯ u k | n dx ≤ ˆ B δ |∇ u k | n dx = τ k := 1 − ˆ R n \ B δ ( |∇ u k | n + | u k | n ) dx − ˆ B δ | u k | n dx = 1 − c − nn − k (cid:18) α lim k →∞ k U k k nn − nα n log δ + A + o k (1) + O δ (1) (cid:19) . (5.1)When x ∈ B Lr k , by (5.1) and Lemma 4.8, we have α k u nn − k ≤ α n (1 + α k u k k nn ) n − (¯ u k + u k ( δ )) nn − ≤ α n | ¯ u k | nn − + nα n n − | ¯ u k | n − | u k ( δ ) | + α n αn − (cid:13)(cid:13)(cid:13)(cid:13) c n − k u k (cid:13)(cid:13)(cid:13)(cid:13) nn + o k (1) ≤ α n | ¯ u k | nn − + nα n n − | c k | n − | u k ( δ ) | + α n αn − k →∞ k U k k nn + o k (1) ≤ α n | ¯ u k | nn − + nα n n − | G α ( δ ) | + α n αn − k →∞ k U k k nn + o k (1)= α n | ¯ u k | nn − − n n − δ + nα n n − A + α n αn − k →∞ k U k k nn + o k (1) + o δ (1) ≤ α n | ¯ u k | nn − τ n − k + α n A − log δ n + o k (1) + o δ (1)Integrating the above estimates on B Lr k , we have ˆ B Lrk (cid:16) exp n α k u nn − k o − (cid:17) dx ≤ δ − n exp { α n A + o k (1) } · sharp Trudinger-Moser type inequality in R n · ˆ B Lrk (cid:18) exp (cid:26) α k u nn − k /τ n − k (cid:27) − (cid:19) dx + o k (1) . By the Lemma 5.1, we have ˆ B Lrk (cid:16) exp n α k u nn − k o − (cid:17) dx ≤ ω n − n exp (cid:26) α n A + 1 + 12 + . . . + 1 n − (cid:27) , thanks to Lemma 4.6, we getlim k →∞ ˆ R n Φ (cid:16) α k u nn − k (cid:17) dx ≤ lim L →∞ lim k →∞ ˆ B Lrk (cid:16) exp (cid:16) α k u nn − k (cid:17) − (cid:17) dx ≤ ω n − n exp (cid:26) α n A + 1 + 12 + . . . + 1 n − (cid:27) . (5.2)Combining (5.2) and Lemma 4.3, the proposition is proved.In this subsection, we will construct a function sequence { u ε } ⊂ W ,n ( R n ) with k u ε k W ,n =1 such that ˆ R n Φ (cid:16) α n u nn − ε (cid:17) dx > ω n − n exp (cid:26) α n A + 1 + 12 + . . . + 1 n − (cid:27) . Proof of Theorem 1.2.
Let u ε = C − C − n − (cid:18) n − αn log (cid:18) c n | xε | nn − (cid:19) − B ε (cid:19)(cid:18) αC − nn − k G α k nn (cid:19) n | x | ≤ Rε, G α ( | x | ) (cid:16) C nn − + α k G α k nn (cid:17) n Rε < | x | , where c n = (cid:0) ω n − n (cid:1) n − , B ε , R and c depending on ε will also be determined later, such thati) Rε → R → ∞ and C → ∞ , as ε → C − n − αn C − n − log (cid:16) c n | R | nn − (cid:17) + B ε (cid:18) αC − nn − k G α k nn (cid:19) n = G α ( Rε ) (cid:16) C nn − + α k G α k nn (cid:17) n ;We can obtain the information of B ε , C and R by normalizating u ε . By Lemma 4.9, wehave ˆ R n \ B Rε ( |∇ u ε | n + | u ε | n ) dx G. Lu and M. Zhu = 1 C nn − + α k G α k nn ˆ R n \ B Rε ( |∇ G α | n + | G α | n ) dx = 1 C nn − + α k G α k nn (cid:18) − G α ( Rε ) ˆ ∂B Rε (cid:18) |∇ G α | n − ∂G α ∂n (cid:19) dx + α ˆ R n \ B Rε | G α | n dx (cid:19) = G α ( Rε ) ω n − | G ′ ( Rε ) | n − ( Rε ) n − + α ´ R n \ B Rε | G α | n dxC nn − + α k G α k nn , and ˆ B Rε ( |∇ u ε | n ) dx = n − α n (cid:16) C nn − + α k G α k nn (cid:17) ˆ c n R nn − u n − (1 + u ) n du = n − α n (cid:16) C nn − + α k G α k nn (cid:17) ˆ c n R nn − ((1 + u ) − n − (1 + u ) n du = n − α n (cid:16) C nn − + α k G α k nn (cid:17) n − X k =0 C kn − ( − n − − k n − k − (cid:16) c n L nn − (cid:17) + O (cid:16) R − nn − (cid:17)(cid:17) , using the fact that E := n − X k =0 C kn − ( − n − − k n − k − − (cid:18) · · · + 1 n − (cid:19) , we have ˆ B Rε ( |∇ u ε | n ) dx = − n − α n (cid:16) C nn − + α k G α k nn (cid:17) (cid:16) E − log (cid:16) c n R nn − (cid:17) + O (cid:16) R − nn − (cid:17)(cid:17) It is easy to check that ˆ B Rε ( | u ε | n ) dx = O (( Rε ) n C n ) , thus we get ˆ R n ( |∇ u ε | n + | u ε | n ) dx = 1 α n (cid:16) C nn − + α k G α k nn (cid:17) (cid:16) ( n − E + ( n −
1) log (cid:16) c n R nn − (cid:17) − log ( Rε ) n + α n A + αα n k G α k nn + O ( φ )where φ = ( Rε ) n C n + ( Rε ) n log n Rε + R − nn − + C − nn − + C n n − R n ε n . sharp Trudinger-Moser type inequality in R n ´ R n ( |∇ u ε | n + | u ε | n ) dx = 1, we have α n (cid:16) C nn − + α k G α k nn (cid:17) = ( n − E + ( n −
1) log (cid:16) c n R nn − (cid:17) − log ( Rε ) n + α n A + αα n k G α k nn + O ( φ ) , that is(5.3) α n C nn − = ( n − E + log ω n − n − log ε n + α n A + O ( φ ) . On the other hand, by ii) we have C − C − n − (cid:18) n − α n log (cid:16) c n | R | nn − (cid:17) − B ε (cid:19) = − nα n log Rε + A + O ( φ ) C n − which implies that(5.4) C nn − = − nα n log ε + log ω n − n − B ε + A + O ( φ ) . Combining (5.3) and (5.4), we have(5.5) B ε = − n − α n E + O ( φ )Setting R = − log ε , which satisfies Rε → ε →
0. We can easily verify that(5.6) k u ε k nn = k G α k nn + O (cid:16) C n n − R n ε n (cid:17) + O ( R n ε n ( − log ( Rε ) n )) C nn − + α k G α k nn . It is well known that when | t | < − t ) nn − ≥ − nn − t and (1 + t ) − n − ≥ − tn − x ∈ B Rε ,0 G. Lu and M. Zhu α n | u ε | nn − (1 + α k u ε k nn ) n − = α n C nn − (cid:16) − C − nn − (cid:16) n − α n log (cid:16) c n (cid:12)(cid:12) xε (cid:12)(cid:12) nn − (cid:17) − B ε (cid:17)(cid:17) nn − (cid:16) αC − nn − k G α k nn (cid:17) n − (1 + α k u ε k nn ) n − ≥ α n C nn − (cid:18) − nn − C − nn − (cid:18) n − α n log (cid:18) c n (cid:12)(cid:12)(cid:12) xε (cid:12)(cid:12)(cid:12) nn − (cid:19) − B ε (cid:19)(cid:19) ·· (cid:16) − αC − nn − k G α k nn (cid:17) n − (1 + α k u ε k nn ) n − ≥ α n C nn − (cid:18) − nn − C − nn − (cid:18) n − α n log (cid:18) c n (cid:12)(cid:12)(cid:12) xε (cid:12)(cid:12)(cid:12) nn − (cid:19) − B ε (cid:19)(cid:19) ·· (cid:16) − αC − nn − k G α k nn (cid:17) n − α k G α k nn + O (cid:16) C n n − R n ε n (cid:17) + O ( R n ε n ( − log ( Rε ) n )) C nn − n − ≥ α n C nn − (cid:18) − nn − C − nn − (cid:18) n − α n log (cid:18) c n (cid:12)(cid:12)(cid:12) xε (cid:12)(cid:12)(cid:12) nn − (cid:19) − B ε (cid:19)(cid:19) ·· (cid:16) − α C − nn − k G α k nn + C − nn − (cid:16) O (cid:16) C n n − R n ε n (cid:17) + O ( R n ε n ( − log ( Rε ) n )) (cid:17)(cid:17) n − ≥ α n C nn − (cid:18) − nn − C − nn − (cid:18) n − α n log (cid:18) c n (cid:12)(cid:12)(cid:12) xε (cid:12)(cid:12)(cid:12) nn − (cid:19) − B ε (cid:19)(cid:19) ·· (cid:18) − n − α C − nn − k G α k nn + C − nn − (cid:16) O (cid:16) C n n − R n ε n (cid:17) + O ( R n ε n ( − log ( Rε ) n )) (cid:17)(cid:19) ≥ α n C nn − − n log (cid:18) c n (cid:12)(cid:12)(cid:12) xε (cid:12)(cid:12)(cid:12) nn − (cid:19) + nα n n − B ε − α n α k G α k nn ( n − C nn − + O ( φ ) . By (5.3) and (5.5), we obtain α n | u ε | nn − (1 + α k u ε k nn ) n − ≥ − E + log ω n − n − log ε n − n log (cid:18) c n (cid:12)(cid:12)(cid:12) xε (cid:12)(cid:12)(cid:12) nn − (cid:19) − α n α k G α k nn ( n − C nn − + α n A + O ( φ ) . Then we have ˆ B Rε Φ (cid:16) α n | u ε | nn − (1 + α k u ε k nn ) n − (cid:17) dx sharp Trudinger-Moser type inequality in R n ≥ exp ( − E + α n A + log ω n − n − log ε n − α n α k G α k nn ( n − C nn − + O ( φ ) ) ·· ˆ B Rε exp (cid:26) − n log (cid:18) c n (cid:12)(cid:12)(cid:12) xε (cid:12)(cid:12)(cid:12) nn − (cid:19)(cid:27) ≥ c n − n ε − n exp ( − E + α n A − α n α k G α k nn ( n − C nn − + O ( φ ) ) ˆ B Rε (cid:18) c n (cid:12)(cid:12)(cid:12) xε (cid:12)(cid:12)(cid:12) nn − (cid:19) − n dx ≥ ( n − ω n − n exp ( − E + α n A − α n α k G α k nn ( n − C nn − + O ( φ ) ) ˆ c n R nn − u n − (1 + u ) n du ≥ ( n − ω n − n exp ( − E + α n A − α n α k G α k nn ( n − C nn − + O ( φ ) ) (cid:18) n − o (cid:16) R − nn − (cid:17)(cid:19) ≥ ω n − exp {− E + α n A } n − α n α k G α k nn ( n − C nn − + O ( φ ) ! On the other hand, we have ˆ R n \ B Rε Φ (cid:16) α n u nn − ε (cid:17) dx ≥ α n − n ( n − C nn − ˆ R n \ B Rε | G α | n dx = α n − n k G α k nn + O ( R n ε n (log ( Rε ) n ))( n − C nn − , thus ˆ R n Φ (cid:16) α n | u ε | nn − (1 + α k u ε k nn ) n − (cid:17) dx ≥ ω n − n exp {− E + α n A } − α n α k G α k nn ( n − C nn − + O ( φ ) ! + α n − n k G α k nn ( n − C nn − . Since R = log ε , by (5.4) we have R ∼ C nn − , then we can easily verify that φ = o (cid:16) C − nn − (cid:17) .Hence when α small enough, we have ˆ R n Φ (cid:16) α n | u ε | nn − (1 + α k u ε k nn ) n − (cid:17) dx > ω n − n exp {− E + α n A } . Therefore the proof of Theorem 1.2 is completely finished.
Acknowledgement:
The main results of this paper were presented by the second authorin the AMS special session on Geometric Inequalities and Nonlinear Partial DifferentialEquations in Las Vegas in April, 2015.2
G. Lu and M. Zhu
References [1] S. Adachi, K. Tanaka, Trudinger type inequalities in R N and their best exponents, Proc.Amer. Math. Soc., (2000), 2051-2057.[2] D.R. Adams, A sharp inequality of J. Moser for higher order derivatives, Ann. of Math.,128 (2) (1988), 385-398.[3] Adimurthi, O. Druet, Blow-up analysis in dimension 2 and a sharp form of Trudinger–Moser inequality, Comm. Partial Differential Equations., (2004), 295–322.[4] Adimurthi, K. Sandeep, A singular Moser–Trudinger embedding and its applications,NoDEA Nonlinear Differential Equations Appl., (5–6) (2007), 585–603.[5] D. M. Cao, Nontrivial solution of semilinear elliptic equation with critical exponent in R , Comm. Partial Differential Equations., (1992), 407-435.[6] L. Carleson, S.-Y. A. Chang, On the existence of an extremal function for an inequalityof J. Moser, Bull. Sci. Math., (1986), 113-127.[7] D.G. de Figueiredo, J.M. do ´o and B. Ruf, Elliptic equations and systems with criticalTrudinger–Moser nonlinearities, Discrete Contin. Dyn. Syst., (2011), 455-476.[8] M. del Pino, M. Musso and B. Ruf, New solutions for Trudinger-Moser critical equationsin R , J. Funct. Anal. ,(2010), 421C457.[9] M. de Souza, J. M. do ´O, A sharp Trudinger-Moser type inequality in R . Trans. Amer.Math. Soc. (2014), 4513-4549.[10] J. M. do ´O, N-Laplacian equations in R N with critical growth, Abstr. Appl. Anal., (1997), 301-315.[11] J. M. do ´O, M. de Souza and E. de Medeiros, An improvement for the Trudinger-Moserinequality and applications, J. Differential Equations, (2014), 1317-1349.[12] J. M. do ´O and Manasss de Souza, A sharp inequality of Trudinger-Moser type andextremal functions in H ,n R n J. Differential Equations (2015), 4062-4101[13] M. Flucher, Extremal functions for the Trudinger-Moser inequality in 2 dimensions,Comment. Math. Helv., (1992), 471-497.[14] N. Lam, G. Lu, Existence and multiplicity of solutions to equations of N -Laplacian typewith critical exponential growth in R N , J. Funct. Analysis, (2012) 1132–1165.[15] Y. X. Li, Moser-Trudinger inequality on compact Riemannian manifolds of dimensiontwo, J. Partial Differential Equations, (2001),163-192. sharp Trudinger-Moser type inequality in R n (2005), No. 5, 618-648.[17] Y. X. Li, B. Ruf, A sharp Trudinger-Moser type inequality for unbounded domains in R n , Indiana Univ. Math. J., (2008), 451–480.[18] K. C. Lin, Extremal functions for Moser’s inequality, Trans. Amer. Math. Soc., (1996), 2663-2671.[19] P.-L. Lions, The concentration-compactness principle in the calculus of variations. Thelimit case, part 1, Rev. Mat. Iberoamericana, (1985), 145-201.[20] G. Lu, Y. Yang, Sharp constant and extremal function for the improved Moser-Trudinger inequality involving L p norm in two dimension, Discrete Contin. Dyn. Syst., (3) (2009), 963-979.[21] G. Lu, Y. Yang, A sharpened Moser-Pohozaev-Trudinger inequality with mean valuezero in R , Nonlinear Analysis, (2009) 2992–3001.[22] A. Malchiodi, L. Martinazzi, Critical points of the MoserCTrudinger functional on adisk, J. Eur. Math. Soc. , 893C908.[23] J. Moser, Sharp form of an inequality by N. Trudinger, Indiana Univ. Maths J., (1971), 1077–1092.[24] B. Ruf, A sharp Trudinger-Moser type inequality for unbounded domains in R , J.Funct. Analysis, (2004), 340-367.[25] J. Serrin, Local behavior of solutions of qusai-linear equations, Acta. Math., (1964),248-302.[26] M. Struwe, Positive solutions of critical semilinear elliptic equations on non-contractibleplanar domains, J. Eur. Math. Soc., (4) (2000), 329–388.[27] P. Tolksdorf, Regularity for a more general class of qusilinear elliptic equations, J.Differential Equations, (1984), 126-150.[28] N. S. Trudinger, On embeddings in to Orlicz spaces and some applications, J. Math.Mech., (1967), 473–484.[29] Y. Yang, A sharp form of Moser–Trudinger inequality in high dimension, J. Funct.Analysis, (2006) 100–126.[30] J. Zhu, Improved Moser-Trudinger inequality involving L p norm in n dimensions, Adv.Nonlinear Stud.,14