Extremal functions for Trudinger–Moser inequalities with nonnegative weights

Using blow-up analysis, the author proves the existence of extremal functions for Trudinger–Moser inequalities with nonnegative weights on bounded Euclidean domains or compact Riemannian surfaces. This extends recent results of Yang (J. Differ. Equ. 258:3161–3193, 2015) and Yang–Zhu (Proc. Am. Math. Soc. 145:3953–3959, 2017).

When the high order eigenvalues are involved, we have a similar result.
Theorem 2 Let be a smooth bounded domain in R 2 , λ l+1 ( ) be the (l + 1)th eigenvalue of the Laplace operator with Dirichlet boundary condition, and h be in C 0 ( ) with h ≥ 0 and h ≡ 0. Then we see that, for any 0 ≤ α < λ l+1 ( ), the supremum can be attained by some u 0 ∈ E ⊥ l ∩ C 1 ( ) satisfying u 0 1,α = 1, where E ⊥ l is defined as in (1.7) and · 1,α defined as in (1.4).
Similar results hold on compact Riemannian surfaces. Denote by ( , g) a compact Riemannian surface without boundary, by ∇ g its gradient operator and by g the Laplace-Beltrami operator, respectively. Let λ 1 ( ) be the first eigenvalue of g . Denote for all u ∈ W 1,2 ( ) with |∇ g u| 2 dxα u 2 dv g ≥ 0. Then we have the following theorem.
Then we have the weak Trudinger-Moser inequality J(u) ≥ -C, where C is a positive constant depending only on ( , g) and α.
If h is strictly positive and J(u) has no minimizer on H = {u ∈ W 1,2 ( ) : u dv g = 0}, Yang and Zhu [10] calculated the infimum of J(u) on H by using the method of blow-up analysis.
One may refer to [11] for earlier results on the functional 1 2 |∇ g u| 2 dv g + 8π u dv g -8π log he u dv g .
Let λ 1 ( ) < λ 2 ( ) < · · · be all distinct eigenvalues of g and E λ i ( ) be the eigenfunction space associated to λ i ( ). For any positive integer l, denote Similar to Theorem 2, we obtain the following.

Theorem 5 Let ( , g) be a compact Riemannian surface without boundary, h be in C
can be attained by some u 0 ∈ E ⊥ l ∩ C 1 ( ) satisfying u 0 1,α = 1 and u 0 dv g = 0.
Existence of extremal functions for Trudinger-Moser inequality can be traced back to Carleson and Chang [12], where the unit ball case was treated. Later contributions in this direction include M. Struwe [13], Flucher [14], Lin [15], Ding-Jost-Li-Wang [11], Adimurthi-Struwe [16], Li [17], Adimurthi-Druet [6], and so on. In our proof, we use the blow-up method. Compared with [1], there are some different key points. First, we derive the different Euler-Lagrange equation on which the analysis is performed. Then we prove that h must be positive at the blow-up point. Hence we use the different scaling when define the maximizing sequences of functions. We also obtain the different upper bound of the subcritical functionals. Finally, when proving the existence of the extremal function, we obtain the different lower bounds for the integrals of test functions constructed in Sects. 2-5. It should be remarked that our analysis on the weight h is essentially different from that of Yang and Zhu [2], where a weak version of Trudinger-Moser inequality was studied.
The rest of the paper is arranged as follows. In Sects. 2 and 3, we prove the main results in the Euclidean case (Theorems 1 and 2). In Sects. 4 and 5, we prove the main results in the Riemannian surface case (Theorems 3 and 5).

Proof of Theorem 1 2.1 The subcritical functionals
In this subsection, using the method in the calculus of variations, we prove the existence of maximizers for the subcritical functionals.

Lemma 6 For any
Proof For 0 < < 4π , we choose a function sequence u j ∈ W 1,2 0 ( ) such that as j → ∞. Then there exists some u ∈ W 1,2 0 ( ) such that up to a subsequence, Using a similar argument in the spirit of the one in [1], we find that he (4π -)u 2 j is bounded in L q ( ) for some q > 1. Then we get he (4π -)u 2 j → he (4π -)u 2 strongly in L 1 ( ). This together with (2.2) immediately yields (2.1). We claim that u 1,α = 1. Otherwise u 1,α < 1. It follows that There is a contradiction between in (2.1) and (2.3). Hence u 1,α = 1.

Moreover, the Euler-Lagrange equation for u is
Using elliptic estimates, we get u ∈ C 1 ( ). Let c = u (x ) = max u . If c is bounded, the existence of the extremal function is trivial by standard elliptic estimates. Thus we assume that c → ∞ and x → x 0 ∈ . A result of Gidas, Ni and Nirenberg on page 223 of [18] implies x 0 / ∈ ∂ .
Using the same argument as the one in step 2 of [1], we get the energy concentration. For the function sequence u , we have u 0 weakly in W 1,2 0 ( ), u → 0 strongly in L q ( ) for any q > 1, and |∇u | 2 dx δ x 0 in the sense of measure as → 0, where δ x 0 denotes the Dirac measure centered at x 0 .
Next we prove that h is positive at the blow-up point x 0 . This property plays an important part in our analysis.
Proof We prove it by contradiction. Suppose that h(x 0 ) = 0. Note that up to a sequence where η is a positive constant. Let be sufficiently small such that where o r (1) → 0 as r → 0.
Choose r sufficiently small such that Here we have used the Trudinger-Moser inequality (1.5).

Blow-up analysis
We shall analyze the behavior of the maximizers by using a blow-up analysis. Let Using the Hölder inequality and the classical Trudinger-Moser inequality, we have where 0 < δ < 4π , C depends only on h and δ. Thus we get We define two sequences of functions on : They satisfy the following equation: It is clear that → R 2 as → 0. Noting that |ψ | ≤ 1 and ψ → 0 uniformly in as → 0 and using the elliptic estimates, we get ψ → ψ in C 1 loc (R 2 ), where ψ is a bounded harmonic function in R 2 . Since ψ(0) = lim →0 ψ (0) = 1, we have by the Liouville theorem Similarly, we have by the elliptic estimates where ϕ satisfies ϕ = e 8πϕ in R 2 and ϕ(0) = 0.
A result of Chen and Li [19] implies that For the convergence behavior away from x 0 , we have c u G weakly in W 1,p 0 ( ) for any 1 < p < 2, strongly in L q ( ) for any q ≥ 1 and in C 1 loc ( \{x 0 }), where G is a Green function satisfying where δ x 0 is the Dirac measure centered at x 0 . G can be represented by where A 0 is a constant depending on x 0 and α, ∈ C 1 ( ) with (x 0 ) = 0.
This together with (2.10) leads to The argument in the proof of Lemma 3.3 in [20] yields

Proof of Theorem 1
Let l be a positive integer and 0 ≤ α < λ l+1 ( ). Following the same steps as in the proof of Theorem 1, we see that, for any , 0 < < 4π , there exists some u ∈ E ⊥ l ∩ C 1 ( ) with u 1,α = 1 such that where · 1,α is defined as in (1.4). Moreover, the Euler-Lagrange equation for u is Let c = |u (x )| = max |u |. We assume that c → ∞ and x → x 0 ∈ . Similar to (2.11), we obtain for r > 2R , as in (2.12). Set where (e ij ) (1 ≤ i ≤ l, 1 ≤ j ≤ n i ) is the basis of E l . Then, by (75) and (76) Then he 4π φ 2 dx > γ + πh(x 0 )e 1+4π A 0 . This contradicts (3.1). Hence c must be bounded and the extremal function exists. We finish the proof of Theorem 2.

Proof of Theorem 3
First, we prove that, for any 0 < < 4π , there exists some u ∈ C 1 ( ) such that with u 1,α = 1 and u dv g = 0.
The main procedure of the proof is as follows. Since 0 ≤ α < λ 1 ( ), we may choose a bounded sequence u j in W 1,2 ( ) such that There exists some u ∈ W 1,2 ( ) such that up to a subsequence, u j u weakly in W 1,2 ( ), u j → u strongly in L 2 ( ), u j → u a.e. in .
Using the same argument as in the proof of Theorem 3 in [1], we get he (4π -)u 2 j is bounded in L q for some q > 1. Hence he (4π -)u 2 j → he (4π -)u 2 strongly in L 1 ( ). Hence (4.1) holds. The fact that u j dv g = 0 implies u dv g = 0. We also have u 1,α = 1 by contradiction as in the proof of Lemma 6.
Moreover, u satisfies the Euler-Lagrange equation where g denotes the Laplace-Beltrami operator.
Denote c = |u (x )| = max |u |. If c is bounded, the existence of the extremal function follows from the elliptic estimates. We assume that c → +∞ and x → p ∈ . Similar to Lemma 7, we have h(p) > 0. Choosing an isothermal coordinate system (U, φ) near p such that the metric g can be written as g = e f (dx 2 1 + dx 2 2 ), where f ∈ C 1 (φ(U), R) and f (0) = 0. Denote = φ(U), u = u • φ -1 and x = φ(x ). Let and Then we get where -R 2 is the usual Laplace operator in R 2 . By the same argument as in Sect. 2.2, we obtain where ϕ(x) = -1 4π log 1 + π|x| 2 and R 2 e 8πϕ dx = 1.
We also have c u G weakly in W 1,q ( ) for all 1 < q < 2, and c u → G in C 1 loc ( \{p}) ∩ L 2 ( ), where G is Green function satisfying and G dv g = 0. As before, G can be represented by where r is the geodesic distance from p, A p is a constant and p ∈ C 1 ( ) with p (p) = 0. Similar to (2.11), we can get sup u∈W 1,2 ( ), u dv g =0, u 1,α ≤1 where γ 1 = h dv g .
Using the elliptic estimates, we have the existence of the extremal function.