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

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.