Asymptotic properties of critical points for subcritical Trudinger-Moser functional

1 Introduction

Let Ω ⊂ R 2 be a smooth bounded domain. It is well-known that there is a Sobolev embedding W 0 1 , p ( Ω ) ↪ L 2 p / ( 2 − p ) ( Ω ) for p ∈ [ 1 , 2 ) . If we look at the limiting Sobolev case p = 2 , then H 0 1 ( Ω ) ≔ W 0 1 , 2 ( Ω ) ↪ L q ( Ω ) for any q ≥ 1 , but H 0 1 ( Ω ) ̸ x ↪ L ∞ ( Ω ) . To fill this gap, it is natural to look for the maximal growth function g : R → R + such that

where ‖ ∇ u ‖ 2 2 = ∫ Ω ∣ ∇ u ∣ 2 d x denotes the Dirichlet norm of u . Pohozaev [20] and Trudinger [23] proved independently that the maximal growth is of exponential type and more precisely that there exists a constant α such that

Later, this inequality was sharpened by Moser [16] as follows:

Lions [15] showed that for (1) there is a loss of compactness at the limiting exponent α = 4 π . But, despite the loss of compactness, the existence of a function which attains the supremum in (1) for α = 4 π is shown by Carleson and Chang [1] if Ω is a unit ball. This result was extended to arbitrary bounded domains in R 2 by Flucher [7].

In the case of the whole space Ω = R 2 , Ruf [21] showed that for α ≤ 4 π

and that d α is attained if α = 4 π . By Li and Ruf [13], the critical Trudinger-Moser inequality in R N ( N ≥ 3 ) and its estremals are established. It is proved by Ishiwata [9] that there exists an explicit constant C R 2 such that d α is attained for C R 2 < α < 4 π , whereas d α is not attained for sufficiently small α , by vanishing loss of compactness, that is, the noncompactness of the Sobolev embedding H rad 1 ( R 2 ) ↪ L 2 ( R 2 ) . By Chen et al. [2,3], the Trudinger-Moser inequalities with a degenerate potential and their higher dimensional case were studied.

In this article, we consider positive critical points of

constrained to the manifold

where λ > 0 is a parameter. By the compactness of E α ∣ Σ λ , i.e., by the continuity of E α with respect to weak convergence sequence in Σ λ , there is a maximizer for sup u ∈ Σ λ E α ( u ) , which is a critical point of E α ∣ Σ λ . Critical points of E α ∣ Σ λ correspond to solutions of the nonlocal problem

where ν is the unit outer normal to ∂ Ω . In addition to maximizers for sup u ∈ Σ λ E α ( u ) , the constant ( λ ∣ Ω ∣ ) − 1 / 2 is also a solution of (3), where ∣ Ω ∣ denotes the Lebesgue measure of Ω . Obviously, u is a solution of (3) if and only if u λ ( x ) = u ( ( x − p ) / λ ) is a solution of

for p ∈ R 2 and Ω λ ≔ { λ x + p ∣ x ∈ Ω } . Thus, the parameter λ means the scaling of the domain. The aim of this article is to study asymptotic behavior of critical points of E α ∣ Σ λ both as λ → 0 and as λ → + ∞ .

In [14,17–19], they considered the following Neumann problem for power-type nonlinearity:

where ε is a parameter and f satisfies some conditions with f ( t ) = O ( t p ) as t → ∞ for p > 1 . In [18], it is shown that the constant solution is the only positive solution for (4) provided that ε is sufficiently large. In the case of small ε , it is proved by [14,17,19] that a solution at this least energy level for the Neumann problem possesses just one local maximum point, which lies on the boundary and concentrates (up to subsequences) around a point where mean curvature maximizes. The method employed consists of a combination of the variational characterization of the solutions and exact estimates of the value of the energy functional based on a precise asymptotic analysis of the solutions.

To state our results, let us define the constant I ( α , λ ) by

for α ∈ ( 0 , 2 π ) and λ > 0 . We make a remark that all maximizers for I ( α , λ ) belong to C 2 , β ( Ω ¯ ) and are strictly positive in Ω ¯ . We also define I α by

where R + 2 ≔ { x ∈ R 2 ∣ x 2 > 0 } is the half space. Then the constant α ∗ is defined by

Note that α ∗ ∈ ( 0 , 2 π ) holds. Indeed, by the radially symmetric rearrangement I α = d 2 α / 2 holds, where d 2 α is defined in (2) for 2 α . Moreover, due to Ishiwata [9], d 2 α > 2 α if α is close to 2 π and d 2 α = 2 α if α is sufficiently small. Thus, I α > α holds if α is close to 2 π and I α = α holds if α is small, which imply that α ∗ ∈ ( 0 , 2 π ) .

The constant α ∗ is the threshold of the existence of a maximizer for I α . The result is proved in Section 4. Thresholds in terms of attainability of best constant such α ∗ also appear in other variational problems related to the Trudinger-Moser inequalities (see for instance [4,8, 10–12]).

In this setting, we obtain the following results:

In the case of α ∈ ( α ∗ , 2 π ) , there is a maximizer for I α . Thus, the situation is similar to the case of power-type nonlinearity (4). For large λ , the maximizer u λ has a unique maximum located on the boundary of the domain, and u λ can be made arbitrarily small in the outer region Ω ¯ ⧹ B R / λ ( x λ ) . In addition to Theorem 1.1, we derive that u λ converges to some maximizer of I α in some sense as λ → + ∞ , and subsequently, lim λ → ∞ I ( α , λ ) = I α . In the case of α ∈ ( 0 , α ∗ ) , I α is not attained by vanishing loss of compactness for maximizing sequences. The situation is completely different from the case of (4). Theorem 1.2 asserts that the vanishing phenomena occur for sequences of maximizers. Furthermore, in the case of α ∈ ( 0 , α ∗ ) , it follows from Theorem 1.2 that lim λ → ∞ I ( α , λ ) = I α .

In the proofs of Theorems 1.1 and 1.2, we use a diffeomorphism straightening a boundary portion around a point on ∂ Ω , which was introduced in [14,17,19]. Moreover, we use some results of the solution of the following equation:

Regarding the equation, it is known that all positive solutions are in C 2 ( R 2 ) and radially symmetric for any L ∈ ( 0 , 1 ) . Moreover, these solutions and their first derivatives decay exponentially at infinity. Ruf and Sani [22], proved that for each L ∈ ( 0 , 1 ) there exists a solution which attains the ground state level. We use these results to reject the possibility that maximizer u λ has infinitely many peaks in Ω ¯ .

The following result is asymptotic behavior of positive critical points of E α ∣ Σ λ as λ → 0 .

Theorem 1.3 implies that v λ / ‖ v λ ‖ L ∞ ( Ω ) → 1 in C 2 ( Ω ¯ ) and ( λ ∣ Ω ∣ ) 1 / 2 ‖ v λ ‖ L ∞ ( Ω ) → 1 as λ → 0 . To prove the theorem, we show that ‖ v λ ‖ L ∞ ( Ω ) → ∞ as λ → 0 and use a blow-up analysis. Different from the result obtained in [18] for equation (4), uniqueness of the positive critical point of E α ∣ Σ λ for small λ is still open.

This article is organized as follows. In Section 2, we prove Theorems 1.1 and 1.2. By using asymptotic analysis, we show that either “concentration at one point” or “vanishing” occurs on sequence of maximizers. To prove this claim, we will investigate the asymptotic behavior of maximizers in the region around concentration point as well as in the outer region. In Section 3, we prove Theorem 1.3. In Section 4, the relationship between d α and I α is discussed. In particular, we show that α ∗ is the threshold that divides existence and non-existence of a maximizer for I α .

2 Proofs of Theorems 1.1 and 1.2

In this section, we prove Theorems 1.1 and 1.2. To derive the asymptotic behavior of u λ , we study a nonlocal elliptic equation and estimate I ( α , λ ) .

Before proving Theorems 1.1 and 1.2, we recall some facts about a diffeomorphism straightening a boundary portion around a point P on ∂ Ω , which was introduced in [14,17,19]. Fix P ∈ ∂ Ω . We may assume that P is the origin and the inner normal to ∂ Ω at P points in the direction of the positive x 2 -axis, where x = ( x 1 , x 2 ) ∈ R 2 . In a neighborhood N of P , ∂ Ω ∩ N can be represented by

where γ is the curvature of ∂ Ω at P . Define a map x = Φ ( y ) = ( Φ 1 ( y ) , Φ 2 ( y ) ) by

Since ψ ′ ( 0 ) = 0 , the differential map D Φ of Φ satisfies D Φ ( 0 ) = I , the identity map. Thus, Φ has the inverse mapping y = Φ − 1 ( x ) for small ∣ x ∣ . We write Ψ ( x ) = ( Ψ 1 ( x ) , Ψ 2 ( x ) ) instead of Φ − 1 ( x ) .

For fixed α ∈ ( 0 , 2 π ) and a sequence λ n such that λ n → ∞ as n → ∞ , a maximizer of I ( α , λ n ) is denoted by u n . The maximizer u n ∈ Σ λ n satisfies