Brezis' first open problem - An elliptic equation involving the critical exponent in 3D

Sat, 18 Apr 2026 00:00:00 +0000

Yamabe problem #

Yamabe problem: Suppose $(\mathcal{M}, g_0)$ is a compact closed Riemannian manifold with dimension $N \geq 3$, does there exist a conformal metric $g = u^{\frac{4}{N-2}}g_0$ which has constant scalar curvature $R_g \equiv C$?

Find $u > 0$ on $\mathcal{M}$ such that $$ -\frac{4(N-1)}{N-2}\Delta_{g_0}u + R_{g_0}u = Cu^{\frac{N+2}{N-2}}\qquad\text{on }\mathcal{M}. $$

Some results:

Trudinger [1968]: if $g$ has non-positive scalar curvature.
Aubin [1976]: $N \geq 6$ and $(\mathcal{M}, g)$ not locally conformally flat.
Schoen [1984]: any dimension, the remaining cases, assuming the Positive Mass Theorem by Schoen-Yau [1979].

A special case #

Consider the special case where $\mathcal{M}$ is a bounded domain $\Omega$ in $\mathbb{R}^{N}$: $$ \begin{cases} -\Delta u = u^{\frac{N+2}{N-2}}\qquad\text{in }\Omega, \\ u > 0\qquad\text{in }\Omega, \\ u = 0\qquad\text{on }\partial\Omega. \end{cases} $$

Pohozaev [1965]: if $\Omega$ is star-shaped, then there is no nontrivial solution.

Brezis-Nirenberg problem #

Consider a lower-order perturbation: $$ \begin{cases} -\Delta u = u^{\frac{N+2}{N-2}} + \lambda u\qquad\text{in }\Omega, \\ u > 0\qquad\text{in }\Omega, \\ u = 0\qquad\text{on }\partial\Omega. \end{cases} $$

Some results:

Pohozaev’s result also yields nonexistence when $\lambda \leq 0$ and $\Omega$ is star-shaped.
If a positive solution exists, then necessarily $\lambda < \lambda_1$, where $\lambda_1$ is the first eigenvalue of $-\Delta$ on $\Omega$ with zero Dirichlet boundary condition.

Hence, for positive solutions on star-shaped domains, $$ 0 < \lambda < \lambda_1. $$

Brezis’ Open Problem 1.1 #

Let $N=3$, and let $\Omega = B_1 \subset \mathbb{R}^3$ be the unit ball. Consider $$ \begin{cases} -\Delta u = u^5 + \lambda u \qquad \text{in } B_1, \\ u = 0 \qquad \text{on } \partial B_1. \end{cases} $$ We ask whether this problem admits a nontrivial positive solution $u \not\equiv 0$.

Here the exponent $5 = \frac{N+2}{N-2}$ is the critical Sobolev exponent when $N=3$, and this is exactly the source of the main compactness difficulty.

Let $\lambda_1$ be the first Dirichlet eigenvalue of $-\Delta$ on $B_1$. The classical Brezis-Nirenberg theory shows:

If $\lambda \leq 0$, then the only solution is $u \equiv 0$.
If $\frac{1}{4}\lambda_1 < \lambda < \lambda_1$, then there exists a positive radial solution.
If $0 < \lambda \leq \frac{1}{4}\lambda_1$, then any radial solution must be trivial; hence there is no positive radial solution.
If $\lambda > \lambda_1$, there exist sign-changing solutions, but no positive solution.

Therefore the unresolved case is:

Open Problem 1.1. Assume $$ 0 < \lambda \leq \frac{1}{4}\lambda_1. $$ Does there exist a nontrivial solution?
Equivalently, since no positive radial solution can exist in this range, can there exist a non-radial positive solution?

This problem has remained open for decades, even if one restricts further to a smaller interval such as $$ 0 < \lambda < \varepsilon $$ for some sufficiently small $\varepsilon > 0$.

Remarks #

A few points are worth emphasizing:

By the Gidas-Ni-Nirenberg symmetry principle, positive solutions on a ball are often expected to be radial; however, in this regime Brezis observed that any radial solution must vanish, so any eventual positive solution would have to be genuinely non-radial.
This makes dimension $3$ sharply different from higher-dimensional cases, where the Brezis-Nirenberg existence theory is better understood.
The bifurcation picture suggests branches of sign-changing non-radial solutions emerging from higher eigenvalues, but it is not known whether such branches can reach the interval $\left(0,\frac14\lambda_1\right]$.

References #

H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math. 36 (1983), 437–477.
H. Brezis, Some of My Favorite Open Problems, Open Problem 1.1.
M. Comte, Solutions of elliptic equations with critical Sobolev exponent in dimension three, Nonlinear Anal. 17 (1991), 445–455.
O. Druet, Elliptic equations with critical Sobolev exponents in dimension 3, Ann. Inst. H. Poincaré Anal. Non Linéaire 19 (2002), 125–142.