Trudinger–Moser Critical Exponential Growth

• Trudinger–Moser Critical Exponential Growth is characterized by the Moser–Trudinger inequality, which ensures that exponential integrals remain finite up to the sharp threshold of 4π.
• Energy concentration and blow-up analysis reveal that solutions often exhibit quantized Dirichlet energies at isolated points, leading to distinctive asymptotic profiles in nonlinear PDEs.
• Extensions to non-Euclidean, fractional, and sign-changing frameworks demonstrate that similar critical thresholds and compactness phenomena apply beyond classical 2D variational problems.

Trudinger–Moser Critical Exponential Growth refers to a threshold phenomenon in nonlinear analysis at which the exponential nonlinearity in two-dimensional variational problems leads to a breakdown of compactness, concentration of energy, and quantization effects. This arises naturally from the sharp embedding of the Sobolev space $H^1_0$ of a two-dimensional domain into Orlicz spaces endowed with exponential-type functionals. The archetype is the Moser–Trudinger inequality, which asserts that for $u \in H^1_0(\Omega)$, $\Omega \subset \mathbb{R}^2$, the exponential integrals $\int_\Omega \exp(\alpha u^2)\,dx$ are finite if and only if $\alpha \le 4\pi$. This threshold $4\pi$ governs the existence, structure, and compactness of critical points for corresponding nonlinear PDEs and variational functionals.

1. Foundations: The Moser–Trudinger Inequality and Critical Growth

Let $B_1$ be the unit disk in $\mathbb{R}^2$. For the Sobolev space $H_0^1(B_1)$ (functions vanishing on the boundary with $\int_{B_1}|\nabla u|^2dx<\infty$), the classical Moser–Trudinger inequality states:

$$\sup_{u\in H_0^1(B_1),\, \int |\nabla u|^2 \leq 1} \int_{B_1} e^{4\pi u^2}\,dx < \infty,$$

where $4\pi$ is sharp: for any $\varepsilon>0$,

$$\sup_{\int |\nabla u|^2 \leq 1} \int_{B_1} e^{(4\pi+\varepsilon)u^2}\,dx = +\infty.$$

The sharp nonlinearity for critical growth is thus $e^{u^2}$, and the associated Moser–Trudinger functional is

$$E(u) = \int_{B_1}(e^{u^2} - 1)dx, \quad u\in H_0^1(B_1).$$

Critical growth means the nonlinear term (e.g., $u e^{u^2}$) grows as fast as permitted by the Moser–Trudinger inequality, precisely at the exponential rate $4\pi$ (Malchiodi et al., 2012).

2. Blow-Up, Energy Quantization, and Loss of Compactness

Positive solutions of elliptic PDEs at critical exponential growth display blow-up and energy quantization phenomena. Consider the constrained variational problem:

$$M_\Lambda = \{u \in H_0^1(B_1): \|\nabla u\|^2 = \Lambda\},$$

and critical points under constraint,

$$-\Delta u = \lambda u e^{u^2},\quad u|_{\partial B_1}=0.$$

If a sequence $u_k$ of positive critical points blows up, i.e. $\max_{B_1} u_k \to \infty$ as $k\to\infty$, then:

• $\Lambda_k \to 4\pi$
• $u_k \rightharpoonup 0$ weakly in $H_0^1(B_1)$,
• $u_k \to 0$ strongly in $C^1_\text{loc}(\overline{B_1}\setminus\{0\})$,
• In measure: $|\nabla u_k|^2dx \rightharpoonup 4\pi \delta_0$ and $$\lambda_k u_k^2 e^{u_k^2}dx \rightharpoonup 4\pi \delta_0$$.

Thus, total Dirichlet energy and "nonlinear energy" both concentrate at a single point (the origin), and the energy quantizes precisely to $4\pi$ (Malchiodi et al., 2012, Druet et al., 2017, Luo et al., 2022).

3. Critical Point Analysis and Radial ODE

Any positive solution to the Euler–Lagrange equation is radially symmetric:

$$- (r u'(r))' = r \lambda u(r) e^{u(r)^2}, \quad u'(0) = 0, \quad u(1) = 0,$$

leading to detailed blow-up analysis:

• Rescale profiles $n_k(x) = u_k(r_k x) - H_k$ around $r=0$ with $H_k = u_k(0)\to\infty$.
• Limiting bubble profile $n_\infty(x) = -\ln(1 + |x|^2)$, solving $-\Delta n_\infty = 4 e^{2 n_\infty}$, energy $4\pi$.
• Precise asymptotics and higher-order corrections can be constructed (Malchiodi et al., 2012, Luo et al., 2022).

4. Existence, Multiplicity, and Nonexistence for Supercritical Energies

• For $\Lambda \leq 4\pi$ (the critical threshold), there exists a maximizer for $E|_{M_\Lambda}$.
• For $\Lambda$ slightly above $4\pi$, one can obtain at least two positive critical points (local maximum and mountain-pass) via variational methods.
• For large enough $\Lambda > \Lambda^* > 4\pi$, there are no positive critical points (nonexistence regime). $\Lambda^*$ depends on domain geometry and is explicit on the disk (Malchiodi et al., 2012, Chen et al., 2022).
• On non-contractible domains (excluding the disk), for every $\Lambda \in (4\pi L, 4\pi(L+1))$, one can construct positive solutions by min–max/topological degree arguments (Marchis et al., 2011).

5. Sign-Changing Solutions, Bubble Clustering, and Breakdown of Quantization

Unlike positive solutions, sign-changing solutions can display non-quantized energy, nonzero weak limits, and clustering of bubbles:

• For symmetric domains, for any $k\geq 1$ and $\beta > 4\pi k$, one can glue $k$ positive bubbles at a single point, with the remainder converging to a sign-changing solution (Martinazzi et al., 2021).
• There is no higher-order energy quantization: total energy can be arbitrary above $4\pi k$, and the weak limit does not vanish, representing much richer blow-up patterns in the sign-changing case.
• In contrast, in the positive case, concentration occurs at isolated points, with energy quantized in $4\pi$-multiples and zero weak limit (Druet et al., 2017, Martinazzi et al., 2021).

6. Uniqueness, Interaction, and Structure of Maximizers

• On the unit disk, any positive solution to $-\Delta u = \lambda u e^{u^2}$ with $0<\lambda<\lambda_1$ is radially symmetric and unique (Chen et al., 2022).
• This uniqueness resolves conjectures about maximizers of the critical Trudinger–Moser inequality.
• For multiple blow-up points, their locations must solve a finite-dimensional interaction system determined by the Green’s function and the domain’s topology, with no bubble towers or mass loss except at isolated quantized points (Druet et al., 2017).
• In hyperbolic balls or domains with conical singularities, quantization and compactness phenomena parallel the Euclidean case but with modified thresholds depending on curvature and cone angles (Chen et al., 19 May 2025, Chen et al., 2022).

7. Higher-Dimensional and Fractional Extensions

• Analogous phenomena hold for the N-Laplacian with critical exponential growth, often replacing the $4\pi$ threshold by the optimal Moser–Trudinger constant $\alpha_N = N\omega_{N-1}^{1/(N-1)}$ (Lam et al., 2010, Araujo et al., 2020, Yang, 2011).
• For fractional Sobolev spaces, there exist fractional Moser–Trudinger inequalities, e.g., for $H^{1/2,2}(\mathbb{R})$, $$\sup_{\|u\|_{1/2}\le 1}\int_{\mathbb{R}}(e^{\pi u^2}-1)dx<\infty$$ (Ó et al., 2018, Ó et al., 2017).
• Cone-type and weighted inequalities adapt the critical exponent to geometry and degeneracy types (Fang et al., 2018).

Summary Table: Sharp Thresholds and Blow-Up Mechanisms

Domain	Critical Threshold	Blow-up Quantization	Compactness
Disk/Ball	$4\pi$	$4\pi \cdot k$	Fails at $4\pi$ and above
Arbitrary $\Omega$	$4\pi$	$4\pi \cdot k$	Topology-dependent
Fractional	$\pi$ ($H^{1/2}$)	$\pi \cdot k$	As above
Cone/Weighted	$4\pi$ (modified)	$4\pi(1+\alpha)$	See (Fang et al., 2018, Chen et al., 19 May 2025)

The phenomenon of Trudinger–Moser critical exponential growth thus constitutes a cornerstone in two-dimensional nonlinear PDE and variational analysis, dictating existence, uniqueness, and asymptotics of solutions at a sharp exponential threshold, with rich geometric, topological, and sign-changing effects. For further details and rigorous proofs, see (Malchiodi et al., 2012, Druet et al., 2017, Chen et al., 2022, Martinazzi et al., 2021, Marchis et al., 2011).